MECHANISM 


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MECHANISM 


BY 
ROBERT  McARDLE  KEOWN,  B.  S, 

ASSOCIATE   PROFESSOR   OF  MACHINE   DESIGN 
UNIVERSITY  OF  WISCONSIN 


FIRST  EDITION 
SIXTH  IMPRESSION 


McGRAW-HILL  BOOK  COMPANY,  INC, 
239  WEST  39TH  STREET.     NEW  YORK 


LONDON:  HILL  PUBLISHING  CO.,  LTD. 
6  &  8  BOUVERIE  ST.,  E.C. 

1912 


Y  THE 
COMPANY 


THE. MAPLE. PRESS- YORK. PA 


PREFACE 

The  writer's  aim  has  been  to  cover  the  subject  of  mechanism 
as  briefly,  simply  and  clearly  as  possible.  The  text  is  designed 
for  a  half  year's  work  of  one  lecture,  one  recitation  and  four 
hours'  drafting  work  per  week. 

No  especial  claim  to  originality  of  subject  matter  can  be  made, 
nor  has  the  writer  found  need  for  a  special  effort  in  this  direction. 
The  arrangement  and  method  of  treatment  are  new,  and  these 
the  author  bases  upon  satisfactory  results  obtained  in  the  work 
of  his  classes  in  the  College  of  Engineering  of  the  University  of 
Wisconsin. 

After  a  brief  discussion  of  motions  and  velocities  linkages  are 
taken  up,  as  they  are  comparatively  easy  for  the  student  to 
understand,  and  simple  problems  can  be  given  out  while  the  sub- 
ject is  yet  new  to  him. 

Cams  are  taken  up  in  detail,  as  they  form  a  part  of  the 
subject  in  which  the  student  needs  considerable  practice  in 
order  to  work  out  original  problems,  and  practically  all  cam 
problems  are  original. 

The  involute  system  of  gearing  is  taken  up  before  the  cycloidal 
system,  because  it  seems  to  the  writer  easier  for  the  student  to 
grasp.  Having  once  become  familiar  with  the  involute  system, 
the  student  can  more  readily  understand  the  cycloidal.  Further- 
more, the  involute  system  is  in  more  general  use  at  the  present 
time. 

Problems  are  given  at  the  end  of  each  chapter.  Some  of  these 
are  designed  especially  for  drafting-room  work,  and  the  necessary 
instructions  as  to.scale,  position  of  drawing  on  the  sheet,  method 
of  procedure  and  time,  etc.,  are  given.  These  few  problems  are 
not  intended  to  exhaust  the  subject,  but  rather  to  serve  as  a 
guide  for  instructors  in  giving  the  drafting-room  work  in  con- 
nection with  the  recitations. 

Quotations  have  been  made  from  the  various  authorities,  and 
credit  given  where  the  quotations  appear.  The  writer  wishes 
to  express  his  thanks  to  the  several  authors  from  whom  he  has 
made  quotations,  the  manufacturing  companies  who  have 
furnished  cuts  and  to  all  others  who  have  aided  in  the  preparation 
of  the  manuscript. 

MADISON,  Wis.,  R-   McA-   KEOWN. 

August,  1912. 

V 


CONTENTS 

PAGE 
PREFACE v 

CHAPTER  I 

MOTIONS  AND  VELOCITIES 1 

Design  of  a  machine — Forms  of  motion — Plane  motion — Rota- 
tion— Translation — Helical  motion — Spherical  motion — Path — 
Velocity — Linear  velocity — Variable  linear  velocity — Angular 
velocity — Relation  between  linear  and  angular  velocity — Problems. 

CHAPTER  II 

INSTANTANEOUS  CENTERS,  KINEMATIC  CHAINS 10 

Mechanisms — Modes  of  connection — Instantaneous  motion — 
Centre — Location  of  the  centro  in  a  single  body — Location  of 
centres  in  three  bodies  moving  relatively  to  each  other — Kine- 
matic chain — Location  and  number  of  centros  in  a  kinematic  chain 
— Problems. 

CHAPTER  III 

SOLUTION  OF  RELATIVE  LINEAR  VELOCITIES  BY  CENTRO  METHOD     .     18 
Relative  linear  velocity — Open   four-linked   mechanisms — Closed 
link  mechanisms — Slider  crank  mechanisms — Problems. 

CHAPTER  IV 

VELOCITY  DIAGRAMS      25 

Linear  velocity  diagrams — Polar  velocity  diagrams — Velocity 
diagram  of  engine  crank  and  crosshead — Variable  motion  mechan- 
isms —  Whitworth  quick-return  motion  mechanism  —  Oscillating 
arm  quick-return  motion  mechanism — Problems. 

CHAPTER  V 

PARALLEL  AND  STRAIGHT-LINE   MOTION   MECHANISMS 37 

Parallelograms — Parallel  ruler — Roberval  balance — Drafting  board 
parallel  mechanisms — Straight-line  motion  mechanisms — Watt's 
straight-line  motion — Scott  Russell  straight-line  motions — 
Thompson  indicator  motion — Tchebicheff's  approximate  straight- 
line  motion — Robert's  approximate  straight-line  motion — Peau- 
cellier's  exact  straight-line  motion — Bricard's  exact  straight-line 
motion — Problems. 

vii 


viii  CONTENTS 

CHAPTER  VI 

PAGE 

CAMS 47 

Contact  between  cam  and  follower — Base  circle — Motions  used  for 
cam  curves — Uniform  motion — Harmonic  motion — Uniformly 
accelerated  motion — Uniformly  retarded  motion — Cams  with  off- 
set followers — Involute  cams — Method  of  laying  out  an  involute 
cam — Cams  with  oscillating  followers— Positive  motion  cams — 
Cams  of  constant  diameter — Main  and  return  cam — Cams  of  con- 
stant breadth — Cylindrical  cams — Cylindrical  cams  with  oscillat- 
ing follower — Inverse  cams — Problems. 

CHAPTER  VII 

GEARING 76 

Friction  gearing — Rolling  cylinders — Velocity  ratio — Grooved 
cylinders — Evan's  friction  cones — Velocity  ratio — Seller's  feed 
disks — Bevel  cones — Brush  plate  and  wheels — Definitions  of 
tooth  parts — Involute  system — Spur  gears — Involute  rack  and 
pinion — Involute  annular  gear  and  pinion — Interference  of  invo- 
lute teeth — Effect  of  Changing  the  distance  between  centers  of 
involute  gears — Interchangeable  involute  gears — Standard  sizes 
of  teeth  for  gears — Laying  out  a  pair  of  standard  involute  gears — 
Cycloidal  system — Cycloid — Epicycloid — Hypocycloid — Applica- 
tion of  cycloidal  curves  to  gear  teeth — Cycloidal  spur  gears — .. 
Cycloidal  rack  and  pinion — Cycloidal  rack  teeth  with  straight 
flanks — Cycloidal  annular  gear  and  pinion — Limiting  size  of  annu- 
lar gear  pinion — Interchangeable  cycloidal  gears — Standard 
diameter  of  rolling  circle  for  cycloidal  gears — Comparison  of  invo- 
lute and  cycloidal  systems — Pin  gearing — Stepped  and  helical 
gears — Approximate  method  of  laying  out  tooth  curves — Cutting 
spur  and  annular  gears — Interchangeable  gear  cutters — Conjugate 
methods  of  cutting  teeth — Fellow's  gear  sharper — Gear  hobbing — 
Problems. 

CHAPTER  VIII 

BEVEL  GEARS,  WORM  AND  WORM  WHEEL 121 

Velocity  ratio — Miter  gears — Crown  gears — Laying  out  the  teeth  of 
bevel  gears— Tredgold's  method — Shop  drawing  of  bevel  gears— 
Cutting  bevel-gear  teeth — Approximate  methods — Accurately  cut 
bevel-gear  teeth — Bilgram  bevel-gear  planer — Worm  and  worm 
wheel — Velocity  ratio — Pitch — Length  of  worm — Cutting  the 
worm-wheel  teeth — Elliptical  gears — Velocity  ratio — Problems. 

CHAPTER  IX 

GEAR  TRAINS 136 

Value  of  a  train — Idle  gear  and  direction  of  rotation — Gear  train 
for  thread  cutting — Problems. 


. 

CONTENTS  ix 

CHAPTER  X 

PAGE 

BELTING 142 

Velocity  ratio  and  directional  relation — Crowning  pulleys — Tight 
and  loose  pulleys — Belts  for  intersecting  shafts — Belts  for  shafts 
that  are  neither  parallel  or  intersecting — Length  of  belts — Stepped 
cones  for  same  length  of  belt — Rope  driving — Chain  driving — 
Renold  silent  chain — Morse  rocker  joint  chain — Problems. 

CHAPTER  XI 

INTEBMITTENT  MOTIONS 155 

Ratchet  gearing — Clutches. 

INDEX ...  .  165 


MECHANISM 

CHAPTER  I 
MOTIONS  AND  VELOCITIES 

The  science  of  Mechanism  treats  of  the  design  and  construction 
of  machinery. 

"A  machine  is  a  combination  of  resistant  bodies  so  arranged 
that  by  their  means  the  mechanical  forces  of  nature  can  be 
compelled  to  produce  some  effect  or  work  accompanied  by 
certain  determinate  motions/'  Reuleaux.  In  general  it  may  be 
said  that  a  machine  is  an  assemblage  of  moving  parts  interposed 
between  the  source  of  power  and  the  work  for  the  purpose  of 
adapting  one  to  the  other. 

1.  Design  of  a  Machine. — In  the  design  of  a  machine  there 
are  three  distinct  parts  to  be  considered: 

First. — The  general  outline  is  sketched  without  any  regard  to 
the  detailed  proportions  of  the  individual  parts,  and  from  this 
sketch  or  skeleton,  by  means  of  pure  geometry  alone,  the  dis- 
placement, velocity,  and  acceleration  of  each  of  the  moving 
parts  can  usually  be  accurately  determined. 

Second. — The  force  acting  on  each  part  has  to  be  determined, 
and  each  part  given  its  proper  form  and  dimensions  to  withstand 
these  forces. 

Third. — Having  designed  the  machine,  the  dynamical  effects 
of  moving  parts  can  be  accurately  determined. 

The  first  part  belongs  to  the  subject  known  as  the  Kinematics 
of  Machines,  the  second  to  the  Design  of  Machine  Parts,  and  the 
third  to  the  Dynamics  of  Machines. 

This  course  will  treat  only  of  the  first  part  dealing  with  the 
motions  of  the  machine  parts,  and  the  manner  of  supporting  and 
guiding  them  without  regard  to  their  strength.  This -is  some- 
times called  Pure  Mechanism  or  the  Geometry  of  Machinery. 

Since  Mechanism  is  a  study  of  relative  motion  it  will  be  well  to 
discuss  the  different  kinds  of  motion. 

2.  Motion. — Motion  is  a  change  of  position.     The  change  of 

1 


'*  MECHANISM 


'position  can  be  noted  only  with  respect  to  the  position  of  some 
other  body  which  is  at  rest,  or  assumed  to  be  at  rest,  or  with 
respect  to  some  body,  the  motion  of  which  is  known  or  assumed 
to  be  known.  That  is,  the  motion  is  purely  a  relative  one. 

Two  bodies  may  be  at  rest  relatively  to  each  other  but  in 
motion  relatively  to  a  third  body,  as  for  example,  two  car  wheels 
fastened  to  the  same  axle  have  no  motions  with  respect  to  each 
other,  but  may  be  in  motion  relatively  to  the  truck  or  track. 

In  problems  dealing  with  machinery  the  motions  of  the  various 
parts  are  usually  taken  with  reference  to  the  frame  of  the  machine. 
This  is  not  always  the  case  however,  as  sometimes  it  is  easier  to 
compare  the  motion  of  one  part  of  a  machine  directly  with  that 
of  some  other  moving  part,  as  for  example  the  number  of  revolu- 
tions that  the  cylinder  of  a  printing  press  makes  to  each  revolu- 
tion of  the  knife  for  cutting  off  the  sheets. 

3.  Forms  of  Motion. — In  order  to  be  of  any  use  in  machine 
construction,  the  motions  must  be  completely  controlled  or  con- 
strained.    Most  of  the  motions  used  are,  or  can  be  reduced  to  one 
of  three  forms,  plane,  helical  and  spherical. 

4.  Plane  Motion. — Plane  motion  is  the  most  common  as  well 
as  being  the  most  simple.     In  order  to  have  plane  motion  all 
points  in  a  plane  section  of  the  body  must  remain  in  that  plane 
and  all  points  outside  that  section  will  move  in  parallel  planes. 

In  Fig.  1,  if  the  lower  face  abed  of  a  cube  is  kept  in  contact  with 


FIG.  1. 

a  flat  table  top,  all  points  in  the  lower  face  will  move  in  a  plane, 
and  all  points  similarly  located  in  any  other  plane  section  will 
move  in  parallel  planes,  no  matter  what  path  the  cube  describes 
in  moving  from  one  position  to  another. 

Thus  if  we  know  the  motions  of  two  points  in  a  body  that  has 
plane  motion,  the  motions  of  other  points  in  the  body  are  also 
known. 


MOTIONS  AND  VELOCITIES  3 

Plane  motion  is  always  found  as  rotation  or  translation,  or  a 
motion  that  can  be  reduced  to  a  combination  of  rotation  and 
translation. 

5.  Rotation. — In  rotation  all  points  in  the  body  move  in  circles, 
that  is,  they  remain  at  a  fixed  distance  from  a  right  line  called 
the  axis  of  rotation.     For  example,  shafts,  pulleys,  fly-wheels, 
etc.,  have  a  motion  of  rotation. 

6.  Translation. — Translation  may  be  divided  into  two  classes, 
Rectilinear  Translation  and  Curvilinear  Translation.     A  body 
has  rectilinear  translation  when  all  points  in  it  move  in  straight 
lines,  as  the  carriage  of  a  lathe,  piston  of  an  engine,  or  spindle  of 
a  drill  press. 

The  crank  pin  of  a  locomotive  driving  wheel  is  an  example  of 
curvilinear  translation.  It  moves  in  a  circle  around  the  center  of 
the  wheel  and  at  the  same  time  moves  along  the  track. 

7.  Helical  Motion. — The  path  traced  by  a  point  moving  at  a 
fixed  distance  from  an  axis  and  with  a  uniform  motion  along  the 
axis  is  a  helix,  and  a  point  moving  in  such  a  path  is  said  to  have 
helical  motion. 

Perhaps  the  most  common  example  of  helical  motion  is  that  of 
a  screw  being  turned  into  a  nut.  All  points  in  the  screw,  except 
those  in  the  axis,  have  helical  motion.  Both  limits  of  helical 
motion  are  plane  motion,  that  is,  if  the  pitch  of  the  screw  is  zero 
the  resulting  motion  is  rotation,  while  if  the  pitch  is  infinity, 
the  motion  will  be  that  of  translation. 

8.  Spherical  Motion. — Spherical  motion  may  be  defined  as  the 
motion  of  a  body  moving  so  that  every  point  in  it  remains  at  a 
constant  distance  from  a  center  of  motion,  but  does  not  remain 
in  a  plane. 

9.  Path. — A  point  in  changing  from  one  position  to  another 
traces  a  line  called  its  path.     The  path  may  be  of  any  form. 
The  path  traced  by  any  point  on  a  pulley  revolving  on  its  shaft 
is  a  circle,  but  a  path  need  not  be  continuous,  as  for  example  the 
path  traced  by  a  rifle  bullet. 

In  general,  the  motion  of  a  body  is  determined  by  the  paths  of 
three  of  its  points  not  in  a  right  line.  If  the  motion  is  in  a  plane, 
two  points  are  sufficient,  and  if  rectilinear,  one  point  determines 
the  motion. 

10.  Velocity. — In  addition  to  knowing  the  path  and  direction 
of  a  moving  body,  there  is  another  element  necessary  to  com- 
pletely determine  its  position,  and  that  is  its  velocity. 


4  MECHANISM 

Heretofore  we  have  not  considered  the  time  necessary  for  a 
body  to  complete  a  certain  motion. 

Velocity  is  measured  by  the  relation  between  the  space  passed 
over  and  the  time  occupied  in  traversing  that  distance.  It  is 
expressed  numerically  by  the  number  of  units  of  distance  passed 
over  in  one  unit  of  time,  as  miles  per  hour,  feet  per  minute,  inches 
per  second,  etc.  It  is  the  rate  of  motion  of  a  point  in  space. 

11.  Linear  Velocity  (V"). — When  a  motion  is  referred  to  a 
point  in  the  path  of  a  body,  its  velocity  is  expressed  in  linear 
measure  and  is  called  its  linear  velocity. 

Velocity  is  uniform  when  equal  spaces  are  passed  over  in  equal 
units  of  time;  that  is,  the  distance  varies  as  the  time. 

Let  V  =  velocity;  S  =  total  distance  passed  over;  r  =  time 
occupied. 

Then  if  we  know  the  distance  passed  over  by  a  body  and  the 
time  it  occupied  in  passing  over  that  distance,  the  velocity, 

S 
F  =— •     Thus  if  a  body  with  a  uniform  velocity  passes  over  a 

distance  of  50  ft.  and  occupies  5  minutes  in  doing  so,  it  has  a 

50 

velocity  of  -=-  =  10  ft.  per  minute, 
o 

If  the  velocity  and  the  time  occupied  are  known,  the  space 
passed  over,  S  =  V  X  T,  while  if  the  space  and  velocity  are  known, 

cr 

the  time  T=-- 

12.  Variable  Linear  Velocity. — A  body  which  has  a  motion 
that  is  accelerated  or  retarded  is  said  to  have  a  variable  linear 
velocity.     Such  a  velocity  is  not  constant,  but  when  calculating 
the  velocity  at  any  point  in  its  path,  its  velocity  is  assumed  to 
be  constant  for  that  instant.     Thus  a  train  starting  from  rest 
and  gradually  increasing  its  speed  until  it  attains  a  velocity  of 
60  miles  per  hour,  has  a  variable  velocity.     After  the  train  had 
gone  a  distance  of  one  mile  from  its  starting-point,  it  might 
have  a  velocity  of  15  miles  per  hour,  meaning  that  if  its  velocity 
remained  constant,  it  would  go  15  miles  in  the  next  hour. 

13.  Angular  Velocity  (7°). — The  angular  velocity  of  a  point 
is  the  number  of  units  of  angular  space  which  would  be  swept 
over  in  a  unit  of  time  by  a  line  joining  the  given  point,  with  a 
point  outside  its  path,  about  which  the  angular  velocity  is  desired. 

The  angular  space  is  measured  in  circular  measure,  or  it  is  the 
ratio  of  the  arc  to  the  radius. 


MOTIONS  AND  VELOCITIES 


5 


The  unit  angle  or  the  radian  is  one  subtended  by  an  arc  equal 
in  length  to  the  radius,  r. 

Hence  in  one  circumference  there  are  —  =  2?r  =  6.2832  radians, 


,. 
oraradian= 


360 


6832  '' 

In  engineering  practice,  angular  velocity  is  usually  expressed 
in  revolutions  per  unit  of  time. 

Thus  if  a  point  S  in  Fig.  2  makes  n  revolutions  per  unit  of 


FIG.  2. 

time,  its  angular,  V°  =  2xn  radians  per  unit  of  time.  This  is 
the  angle  that  a  line  joining  S  with  the  center  of  the  pulley 
will  pass  through  per  unit  of  time. 

14.  All  points  in  the  pulley  of  Fig.  2  have  the  same  angular 
velocity,  since  they  all  pass  through  the  same  angle  in  a  unit 
of  time. 

Also  all  points  in  the  pulley  having  the  same  radii  will  have 
the  same  linear  velocity,  but  the  greater  the  radius  the  larger 
the  linear  velocity.  Thus  if  the  point  M  on  the  pulley  has  a 
radius  r,  the  V~M  =  27rrn  while  the  V~S  =  2nRn.  V~S  :  V~M  = 
2nRn\2nrn 


or 


V-S     

V~M     2nrn 


2-rtRn      R 
r 


Hence  we  may  state  the  following  rule: 

The  linear  velocities  of  all  points  in  the  same  body  are  directly 
proportional  to  their  distances  from  the  center  of  rotation  of  the 
body. 

15.  Relation  between  Linear  Velocity  and  Angular  Velocity. — 
The  angular  velocity  of  a  point  in  a  rotating  body  is  2nn  radians 


6 


MECHANISM 


per  unit  of  time  and  the  linear  velocity  of  a  point  in  the  same  body 
is  2nrn  or  the  angular  velocity  is  the  same  as  the  linear  velocity 
of  a  point  in  the  body  having  a  unit  radius. 

™  ,,  ,,       , ,  .          ,     .,       linear  velocity 

We  may  then  say  that  the  angular  velocity  = 


radius 


v- 


or  V°  —  — .     This  gives  the  angular  velocity  in  radians  and  to 

reduce  it  to  revolutions,  divide  by  2n. 

This  relation  is  often  used  and  should  be  remembered.  The 
angular  velocities  of  two  points  in  different  bodies  having  the  same 
linear  velocities,  but  different  radial  distances  from  their  centers  of 
rotation,  are  inversely  proportional  to  their  radii. 

Let  P  and  S,  Fig.  3,  be  the  two  points  in  A  and  B  respectively 


FIG.  3. 

in  which  R  is  the  radius  of  P  from  its  center  of  rotation  and  r 
the  radius  of  S  from  its  center  of  rotation. 

Let  V-P  =  V~S. 


R  J 
Then  V°A:  V°B 


v-p  v-s 

r 


R 

v-p 


or 


V°A       R 


V°B     V~S 


V°B~  R 


MOTIONS  AND  VELOCITIES  7 

PROBLEMS 

•  1.  An  engine  piston  makes  400  single  strokes  per  minute;  the  flywheel  is 
on  the  crank  shaft.  What  is  the  linear  velocity  of  the  crank  pin  if  the 
length  of  the  crank  is  15  in.? 

«  2.  A  pulley  makes  400  r.p.m.  and  a  point  on  its  rim  has  a  linear  velocity 
of  4000  ft.  per  minute.  How  far  is  the  point  from  the  center  of  rotation? 

3.  A  flywheel  28  ft.  diameter  makes  30  r.p.m.     On  the  flywheel  shaft  is 
a  crank  of  15  in.  radius  which  transmits  motion  to  a  crosshead  by  means  of 
a  connecting  rod  6  ft.  long.     Through  how  many  feet  does  the  crosshead 
travel  per  minute? 

4.  Two  wheels  A  and  B  are  in  contact  and  roll  together  without  slipping. 
The  distance  between  their  centers  is  30  in.     A  makes  80  r.p.m.  and  B 
300  r.p.m.     What  are  their  diameters? 

5.  In  two  wheels  A  and  B  which  roll  together  without  slipping,  their 
diameters  are  16  in.  and  12  in.  respectively,  and  B  makes  125  r.p.m.     What 
are  the  r.p.m.  of  A? 

6.  Velocity  ratio  to  be  transmitted  is  as  3 :4.     Diameter  of  driven  20  in. 
Find  the  diameter  of  the  driver  and  the  distance  between  the  center  of  two 
wheels. 

7.  Plot  the  path  of  D  for  one  complete  revolution  of  the  crank  EC. 
Data :    AE  =  6" ;  BC  =  6i"  ;BD  =  CD  =  5|" ;  AB  =  2Ty ' ;  EC  =  1ft". 

.  Diameter  of  hubs  A  and  E  =  \" 
Diameter  of  hubs  B,  C  and  D  =  f  " 
Diameter  of  pins  A  and  E  =  \" 
Diameter  of  pins  B,  C  and  D=^$" 


r 


Note. — Make  an  ink  drawing,  full  size,  placing  the  point  A  as  indicated. 
Divide  the  circle,  the  radius  of  which  is  EC,  into  12  equal  parts,  numbering 
them  consecutively,  and  also  number  the  corresponding  points  on  the  path 
of  D. 

The  linkage  is  to  be  drawn  in  black  and  other  lines  in  red.  Put  no  dimen- 
sions on  the  drawing. 

The  statement  of  the  problem  and  data  are  to  be  placed  in  the  upper 
right-hand  corner  of  the  sheet.  Time,  4  hours. 

8.  In  the  hand-operated  oil  switch  shown,  through  what  angle  must  the 
handle  AC  move  in  order  to  completely  open  or  close  the  switch? 


MECHANISM 

Data:    Links  AB  EC  BD  DE  EF  FG  GH  HI  HJ  JK 

Length  3i"  8"     6f"  3£"  3"     4£"  4"     2"  4i"  2|" 

Hubs  A    B      D     E      F      G      I      J  K 

Diameter  J"  f"     f"     1"     f"     f"     1"     f"  f" 

Pins,  Diameter  i"  A"  A"  i"    A"  A"  i"    A"  A" 


1JBB  * 

u-TF5      ^*f~ 

N  Border  LineA 


Note. — Make  a  full  size  ink  drawing  and  leave  off  all  dimensions.  Draw 
the  links  in  one  extreme  position,  and  represent  the  other  extreme  by 
center  lines  only.  Time,  6  hours. 

9.  In  the  Tabor  engine  indicator  plot  the  path,  of  B  for  a  motion  of  3J  in. 
for  A  along  the  straight  line  ab. 

Data:     Diameter    of    roll    at    B=A";    AD  =  3$";    CZ>  =  f";    B(7  =  f"; 


Note. — Make  a  pencil  skeleton  drawing,  scale  4  in.  =  1  in.  Take  points 
on  line  ab  $  in.  apart  and  find'  the  corresponding  positions  of  B  and  F. 
Time,  4  hours. 

10.  Design  the  circuit-breaker  so  that  it  will  make  or  break  contact  for  a 
difference  of  13£  in.  in  the  water  level  in  the  tank.  Locate  the  extreme 
positions  of  both  balls  and  of  all  the  levers.  Marble  slab  for  circuit-breaker 
12  in.  square. 

Note. — Make  an  ink  drawing  half  size,  and  design  the  different  parts 
without  regard  to  strength  but  so  that  they  will  look  of  the  proper  pro- 
portions. The  tank  need  not  be  drawn  to  scale,  although  the  difference 


MOTIONS  AND  VELOCITIES 


9 


in  water  level  must  be  drawn  to  the  same  scale  as  the  switch  mechanism. 
Do  not  show  the  rheostat,  motor  or  pump. 

The  statement  of  the  problem  to  be  placed  in  the  upper  right-hand 
corner  of  the  sheet  and  underneath  the  drawing,  AUTOMATIC  FLOAT  SWITCH. 
Time,  6  hours. 


- 


CHAPTER  II 
INSTANTANEOUS  CENTERS,  KINEMATIC  CHAINS. 

16.  Mechanisms. — A  mechanism  or  train  of  mechanism  is  the 
term  applied  to  a  portion  of  a  machine  where  two  or  more  parts 
are  combined  so  that  the  motion  of  the  first  compels  the  motion 
of  the  others  according  to  a  law  depending  on  the  nature  of  the 
combination.     The  two  parts  connected  together  are  known  as 
an  elementary  combination,  so  that  a  train  of  mechanism  consists 
of  a  series  of  elementary  combinations. 

If  a  part  is  considered  separately  from  the  others  it  is  at  liberty 
to  move  in  the  two  opposite  directions  and  with  any  velocity,  as 
the  crosshead  of  an  engine  that  is  not  connected  to  the  connecting 
rod. 

Wheels,  shafts  and  rotating  parts  generally  are  so  connected 
with  the  frame  of  the  machine  that  any  given  point  is  compelled 
when  in  motion,  to  describe  a  circle  around  the  axis,  and  in  a 
plane  perpendicular  to  it.  Sliding  parts  are  compelled  by  fixed 
guides  to  describe  straight  lines,  other  parts  to  move  so  that 
points  in  them  describe  more  complicated  paths  and  so  on. 

These  parts  are  connected  in  successive  order  in  various  ways 
so  that  when  the  first  part  in  the  series  is  moved,  it  compels  the 
second  to  move,  which  again  gives  motion  to  the  third,  etc.  The 
various  laws  of  motion  of  the  different  parts  of  a  train  are  affected 
by  the  mode  of  connection. 

17.  Modes  of  Connection. — Connection  between  the  different 
parts  of  a  machine  may  be  made  in  any  of  the  following  ways: 

a.  Turning  pairs — connected  links. 

b.  Slide  connector — crosshead. 

-     -D  c.   Cams  without  rollers. 

1.  By  direct  contact      '.,«*,.  ,  -  A.          ,-    •, 

d.  Friction  contact — friction  cylinders. 

e.  Rolling  and  sliding  contact — toothed 

gears. 

a.  Cams  with  rollers. 

b.  Rigid  links. 

c.  Flexible  connectors — ropes, 

belts  and  chains. 


2.  By  intermediate  connectors 


10 


INSTANTANEOUS  CENTERS 


11 


a.  Electricity  and  magnetism. 
3.  Without  material  connectors  \  b.  Gravity. 

c.   Centrifugal  force — governor. 

The  first  two  methods  are  perhaps  the  most  important  in  this 
subject,  and  will  be  discussed  later  on. 

18.  Instantaneous  Motion. — When  a  body  changes  its  position 
its  motion  at  any  instant  may  be  said  to  be  its  instantaneous 
motion. 

As  an  illustration  take  the  car  wheel  A}  moving  along  the  track 
B,  Fig.  4.  For  an  instant  there  is  contact  between  the  wheel  and 


FIG.  4. 

the  track  along  a  line  at  C  perpendicular  to  the  paper  and  parallel 
to  the  axis  of  the  wheel.  The  wheel  then  rotates  about  C  which 
becomes  an  element  of  both  the  wheel  and  the  track  for  an 
instant,  and  this  motion  of 'the  wheel  is  its  instantaneous  motion. 
The  line  through  C  and  perpendicular  to  the  paper  is  its  in- 
stantaneous axis. 

If  instead  of  considering  the  whole  wheel,  we  take  a  section 
made  by  passing  a  plane  through  it  perpendicular  to  the  axis,  it 
will  cut  the  line  through  C  in  a  point  which  is  called  the  in- 
stantaneous center  or  centro. 

19.  The  relative  motion  between  A  and  B  is  the  same  whether 
we  consider  the  track  stationary  and  the  wheel  revolving  about 
it,  or  whether  the  wheel  is  considered  stationary  and  the  track 
revolving. 

20.  Centro. — A  centro  is  a  point  common  to  two  bodies  having 
the  same  linear  velocity  in  each,  or  it  is  a  point  in  one  body  about 
which  the  other  tends  to  rotate. 

As  an  illustration  of  the  first  part  of  the  definition,  the  tangent 
point  of  the  two  wheels  A  and  B,  Fig.  3,  is  an  example.  It  is  a 
point  common  to  the  bodies  A  and  B  and  having  the  same  linear 
velocity  in  each.  The  second  part  of  the  definition  is  illustrated 


12  MECHANISM 

by  the  points  in  the  frame  C  of  Fig.  3,  about  which  A  and  B 
rotate. 

21.  Location  of  the  Centre  in  a  Single  Body. — In  any  rigid 
body  moving  about  an  axis,  the  direction  of  motion  of  any  point 
is  perpendicular  to  a  line  joining  the  point  with  the  axis;  con- 
versely, the  axis  of  rotation  will  intersect  a  line  drawn  perpendicu- 
lar to  the  motion  of  any  point  in  the  body,  and  lying  in  the  plane 
of  the  motion  of  the  point. 

An  illustration  of  a  special  case  of  this  statement  is  the  motion 
of  a  point  on  the  rim  of  a  fly  wheel.  Its  instantaneous  direction 
of  motion  is  tangent  to  the  rim,  or  perpendicular  to  a  line  joining 
the  point  with  the  axis  of  rotation. 

A  general  illustration  is  as  follows:  Let  the  points  M  and  N, 
Fig.  5,  have  the  directions  of  motion  shown  by  the  arrows,  both 


FIG.  5. 

motions  lying  in  the  same  plane.  Lines  drawn  through  M  and 
N  and  perpendicular  to  their  respective  directions  of  motion, 
will,  if  they  intersect,  locate  the  centro,  or  instantaneous  center 
of  the  two  points,  and  of  the  body  as  a  whole.  In  Fig.  5  these 
lines  intersect  at  0.  To  determine  the  direction  of  motion  of 
any  other  point  as  P,  draw  a  line  through  P,  perpendicular  to 
the  line  PO. 

In  some  cases  the  centro  cannot  be  located,  as  when  the  lines 
drawn  perpendicular  to  the  directions  of  motion  coincide  or  when 
they  are  parallel.  In  the  latter  case  the  centro  is  at  an  infinite 
distance  away  and  the  body  has  a  motion  of  translation. 

22.  Location  of  Centros  in  Three  Bodies  Moving  Relatively  to 
Each  Other. — Any  three  bodies  moving  relatively  to  each  other 


INSTANTANEOUS  CENTERS 


13 


have  but  three  centres,  and  these  centros  lie  on  the  same  straight 
line. 

In  Fig.  6,  let  A,  B  and  C  be  the  three  bodies  that  move  rela- 
tively to  each  other.  Assume  that  the  body  C  is  held  stationary, 
and  that  A  and  B  are  fastened  by  pin  joints  to  C  at  ac  and  be 
respectively. 

The  number  and  names  of  the  centros  are  found  by  taking  the 
bodies  in  combinations  of  two  using  each  one  with  each  of  the 


FIG.  6. 

others,  as  A  B,  AC  and  BC.  The  bodies  are  usually  represented 
by  capital  letters  and  the  centros  by  lower  case  letters,  so  that 
the  centros  will  be  ab,  ac  and  be.  It  makes  no  difference  in  which 
order  the  letters  are  taken  as  ab  or  ba  for  the  relative  motion  of 
A  about  B  is  the  same  as  the  relative  motion  of  B  about  A 
(Art.  19). 

The  only  motion  that  A  can  have  is  that  of  rotation  about  ac, 
while  the  only  motion  of  B  is  that  of  rotation  about  be.  This 
locates  two  of  the  centros,  and  the  third  one  ab  is  common  to  the 
bodies  A  and  B  and  has  the  same  linear  velocity  in  each. 

Assume  that  ab  lies  at  the  point  o.  When  o  is  considered  as 
a  point  in  A,  it  has  a  radius  from  its  center  of  rotation  o-ac,  and 
moves  in  the  arc  1-2.  Its  direction  of  motion  is  perpendicular 
to  its  radius,  or  along  the  line  m— n. 

When  o  is  considered  as  a  point  in  B}  it  has  a  radius  o-bc  and 


14 


MECHANISM 


moves  in  the  arc  3-4.  Its  direction  of  motion  is  perpendicular 
to  its  radius,  or  along  the  line  p-q. 

Thus  we  have  the  point  o  moving  in  two  different  directions 
at  the  same  time,  which  is  impossible.  The  lines  m-n  and  p-q 
must  be  parallel  or  they  must  coincide.  They  cannot  be  parallel 
since  they  must  both  pass  through  the  same  point,  hence  they 
must  coincide,  and  the  only  time  that  they  will  coincide  is  when 
the  radii  o—ac  and  o-bc  lie  in  the  same  straight  line. 

23.  Kinematic  Chain. — A  kinematic  chain  is  a  combination  of 
rigid  bodies  or  links  so  connected  that  the  motion  of  each  is 
completely  controlled  by  and  depends  upon  the  motions  and 
positions  of  each  of  the  others.  Figs.  7,  8  and  9  are  examples. 


FIG.  7. 


FIG.  8. 


FIG.  9. 

By  holding  one  of  the  links  stationary,  any  motion  given  to  one 
will  cause  a  certain  definite  motion  in  each  of  the  others.  The 
links  can  be  of  any  shape  whatever  as  long  as  their  shape  does 
not  cause  them  to  interfere  with  the  motions  of  any  of  the  other 
links. 


INSTANTANEOUS  CENTERS  15 

Fig.  7  is  a  simple  four-linked  kinematic  chain  consisting  of 
four  turning  pairs,  while  Fig.  8  is  a  simple  chain  of  three  turning 
and  one  sliding  pair.  Fig.  9  is  a  compound  chain.  A  compound 
kinematic  chain  is  one  in  which  one  or  more  of  the  links  has  more 
than  two  joints.  In  Fig.  9  the  links  B  and  E  have  three  joints 
or  are  each  connected  to  three  other  links. 

24.  Location  and  Number  of  Centres  in  a  Kinematic  Chain.— 
To  find  the  centros  in  a  kinematic  chain  write  the  names  of  the 
links  in  a  row,  and  underneath  each,  its  combinations  with  each 
of  the  others.  The  number  of  centros  in  any  kinematic  chain 

N(N—  1) 
=  — ^-— where  N  is  the  number  of  links  in  the  chain. 

2i  > 

In  the  four-link  chain  of  Fig.  10,  the  number  of  centros  will  be 


FIG.  10. 

4(4—1) 

-^•g — -  =6  centros.     Their  names  can  be  found  as  stated  above. 

Zi 

A  B    C    D 

ab  be    cd 

ac  bd 
ad 

Four  of  these  centros,  ab,  be,  cd  and  ad,  can  be  found  at  the 
joints  of  the  links,  leaving  ac  and  bd  to  be  located.  To  locate  the 
position  of  ac  we  know  that  the  three  centros  of  any  three  bodies 
moving  relatively  to  each  other  lie  on  the  same  straight  line 
(Art.  22) ,  and  also  that  to  get  ac  the  links  A  and  C  must  be  used 
since  the  centro  ac  is  common  to  links  A  and  C.  Take  A  and  C 
with  one  of  the  other  links,  say  B.  Then  in  the  links  A,  B,  C,  the 
centros  will  be  ab,  ac,  be.  We  have  already  located  ab  and  bc9 


16 


MECHANISM 


and  since  ac  also  lies  on  the  same  straight  line  with  them,  we  can 
draw  a  line  of  indefinite  length  through  ab  and  be. 

Now  take  with  the  links  A  and  C,  with  a  link  not  used  before, 
as  D.  The  centres  of  A}  C,  D,  will  be  ac,  ad,  cd.  We  already 
know  the  positions  of  ad  and  cd,  and  ac  will  lie  on  the  same 
straight  line  with  them.  Draw  a  line  of  indefinite  length  through 
ad  and  cd.  Now  since  ac  lies  on  this  line,  and  also  on  the  line 
drawn  through  ab  and  be,  it  must  lie  at  the  intersection  of  these 
lines. 

The  centro  bd  is  found  in  a  similar  manner. 

When  there  is  sliding  contact  between  two  links  of  a  kinematic 
chain  as  between  A  and  B,  Fig.  11,  the  centro  will  be  at  an  infinite 


distance  away,  since  it  is  the  point  in  A  about  which  B  tends  to 
rotate,  and  as  B  has  a  motion  of  translation,  the  point  about 
which  B  tends  to  rotate  is  an  infinite  distance  away,  along  a  line 
perpendicular  to  the  direction  of  motion  (Art.  21). 

The  centres  ad,  cd  and  be  being  at  the  joints  of  the  links  are 
readily  found,  and  bd  is  at  the  intersection  of  the  lines  bc-cd  and 
ad-ab. 

The  location  of  the  centros  in  a  compound  chain  can  be  found 
in  a  similar  way.  First  locate  all  of  those  at  the  joints  of  the 
links  and  at  infinity,  and  then  taking  any  three  of  the  links  in 
which  two  of  the  centros  are  known,  find  the  third  one. 


INSTANTANEOUS  CENTERS  17 

.     PROBLEMS 

11.  Prove  that  in  three  bodies  moving  relatively  to  each  other,  their 
three  centros  lie  on  the  same  straight  line. 

12.  Assume  the  positions  of  the  links  in  a  simple  crossed-link  chain  and 
locate  all  of  the  centros. 

13.  Locate  all  of  the  centros  in  Fig.  29  for  the  position  of  the  links  shown. 

14.  Locate  all  of  the  centros  in  Fig.  9  for  the  position  of  the  links  shown. 


CHAPTER  III 

SOLUTION    OF   RELATIVE    LINEAR   VELOCITIES    BY  CENTRO 

METHOD 

25.  Relative  Linear  Velocity. — In  comparing  the  linear  velocity 
of  any  point  in  one  link  with  the  linear  velocity  of  a  point  in  any 
other  link,  the  comparison  must  be  made  through  a  point  com- 
mon to  both  of  them. 

In  Fig.  12  let  A  and  B  be  two  bodies,  held  in  contact  with  each 


FIG.  12. 


other  by  the  frame  C,  which  consider  stationary.  The  centros  of 
A,  B  and  C  are  ab,  ac  and  be.  If  C  is  stationary  the  centros  ac 
and  be  will  be  stationary  since  they  are  the  points  in  C  about 
which  A  and  B  respectively  tend  to  rotate;  and  ab  is  a  point 
common  to  A  and  B  having  the  same  linear  velocity  in  each. 

Let  J  be  a  point  on  A  whose  radius  and  linear  velocity  are 
known,  and  let  K  be  a  point  on  B}  whose  radius  is  known  and 
whose  velocity  it  is  desired  to  find.  The  point  /  has  a  radius 
J-ac  and  revolving  /  about  its  center  will  not  affect  its  velocity 
as  long  as  its  radius  is  not  changed.  Revolve  J  around  to  the 
line  of  centers  and  lay  off  a  line  m-n  to  any  convenient  scale  to 
represent  its  linear  velocity. 

18 


RELATIVE  LINEAR  VELOCITIES  19 

The  velocities  of  all  points  in  the  same  link  or  body  are  directly 
proportional  to  their  distances  from  the  center  of  rotation  (Art. 
14),  so  that  the  linear  velocity  of  ab  having  a  radius  ac-ab  is 
directly  proportional  to  the  linear  velocity  of  J,  having  the  radius 
J-ac.  The  velocity  of  ab  can  now  be  found  by  similar  triangles 
and  is  represented  by  the  line  ab-o. 

Since  the  point  ab  is  common  to  the  bodies  A  and  B,  its  linear 
velocity  will  be  the  same  in  each,  although  its  radius,  when  con- 
sidered as  a  point  in  B  is  not  the  same  as  it  was  when  considered 
as  a  point  in  A.  The  linear  velocities  of  K  and  ab  are  directly 
proportional  to  their  radii. 

Complete  the  triangle  bc-ab-o  by  drawing  the  line  bc-o,  and 
revolve  K  around  its  center  to  the  line  of  centers  at  q.  Draw 
through  q  the  line  q-s,  parallel  to  ab-o.  Then  from  similar 
triangles,  if  ab-o  is  the  linear  velocity  of  ab,  q-s  to  the  same  scale 
will  be  the  linear  velocity  of  K,  or  if  m-n  is  the  linear  velocity  of 
J  then  q-s  is  the  linear  velocity  of  K  to  the  same  scale.  \' 


cd 


FIG.  13. 

26.  Bear  in  mind  that  from  the  linear  velocity  of  the  given 
point,  the  linear  velocity  of  the  common  point  must  first  be  found 
and  from  that  find  the  linear  velocity  of  the  required  point* 

27.  Open  Four  Linked  Mechanisms.     Velocities  of  Points  in 
Adjacent  Links. — Let  A}  B}  C,  D,  Fig.  13,  be  an  open  four-linked 
mechanism  in  which  it  is  required  to  find  the  V~K,  a  point  in  the 
link  C,  knowing  the  V~J,  a  point  in  the  link  B,  for  the  positions 
of  the  links  shown  and  with  the  link  A  stationary. 


20  MECHANISM 

It  is  only  necessary  to  consider  the  links  containing  the  points 
and  the  stationary  link.  These  are  A,  B  and  C.  The  centres  are 
ab,  ac  and  be.  Since  the  link  A  is  stationary,  the  centres  ab  and 
ac  will  be  stationary  and  be  will  be  the  common  point,  or  the 
point  having  the  same  linear  velocity  in  B  and  C. 

The  stationary  centro  ab  is  the  point  in  A  about  which  B 
rotates,  and  since  J  is  a  point  in  B,  it  also  rotates  about  the 
centro  ab. 

Lay  off  from  J  a  length  of  line  to  any  convenient  scale  to 
represent  the  V~J.  (This  line  should  be  laid  off  perpendicular 
to  the  radius  of  J.)  % 

From  the  V~J  find  the  V~bc  by  similar  triangles.  The  linear 
velocities  of  points  in  the  same  link  are  directly  proportional  to 
their  instantaneous  radii  as  well  as  their  actual  radii. 

Revolve  K  about  ac  into  the  line  of  centers  and  after  complet- 
ing the  triangle  of  which  the  V~bc  isf  one  side,  lay  off  the  V~K. 

28.  Velocities  of  Points  in  Opposite  Links. — Given  the  V~J 
Fig.  14  to  find  the  V~K  with  the  link  A  stationary. 


bd 

FIG.  14 

The  links  that  must  be  considered  are  the  two  links  containing 
the  points  and  the  stationary  link,  which  in  this  case  are  A,  B 
and  D,  having  centres  ab,  ad  and  bd.  Since  the  link  A  is  sta- 
tionary, the  centres  of  A  will  be  stationary,  viz.,  ab  and  ad}  and 
bd  is  the  common  point,  or  the  point  common  to  B  and  D,  and 
having  the  same  linear  velocity  in  each. 

Revolve  J  around  into  the  line  of  centers  and  lay  off  a  line  to 
any  convenient  scale  to  represent  its  velocity,  then  by  means  of 
similar  triangles  find  the  linear  velocity  of  the  common  j^oint  bd. 
From  this,  find  the  linear  velocity  of  K,  after  revolving  it  around 
into  the  line  of  centers. 

29.  Crossed  Link  Mechanisms.  Velocities  of  Points  in  Adja- 
cent Links. — Let  it  be  required  to  find  the  V~K,  Fig.  15,  having 
given  the  V~J  with  the  link  B  stationary. 


RELATIVE  LINEAR  VELOCITIES  21 

The  links  to  be  taken  into  consideration  are  Ay  B  and  C, 
having  centres  ab,  ac  and  be,  of  which  ab  and  be  are  stationary. 

Lay  off  a  line  to  represent  the  V~J,  and  by  means  of  similar 
triangles  find  the  V~ac,  the  common  point.  Then  revolve  K 


FIG.  15. 

about  be  into  the  line  of  centers  and  by  means  of  another  set  ot 
similar  triangles  find  the  V~K  as  shown. 

30.  Velocities  of  Points  in  Non-adjacent  Links.— Let  A,  B,  C 
and  Dj  Fig.  16,  be  a  crossed  link  mechanism  in  which  it  is  desired 
to  find  the  V~K,  a  point  in  the  link  D,  having  given  the  V~J  a 
point  in  the  link  B,  with  the  link  A  stationary. 


FIG.  16. 

The  links  that  need  to  be  considered  are  A}  B  and  D,  having 
centros  ab,  ad,  and  bd,  in  which  ab  and  ad  are  stationary  and  bd 
is  the  common  point.  Notice  that  bd  is  located  at  the  intersec- 
tion of  the  links  A  and  C,  but  of  course  there  is  no  joint  at  that 
point,  for  if  there  were,  the  mechanism  would  be  as  one  rigid 
link,  as  there  could  be  no  motion  of  any  of  the  links. 


22 


MECHANISM 


The  link  B  rotates  about  the  centro  ab,  and  as  J  is  a  point  in 
the  link  B,  it  will  rotate  about  ab. 

Revolve  J  into  the  line  of  centers  and  lay  off  a  line  to  represent 
its  linear  velocity.  From  the  V~J,  find  the  V~bd  by  similar 
triangles.  K  rotates  about  the  centro  ad  as  ad  is  the  point  in  A 
about  which  D  rotates.  Revolve  K  around  ad  into  the  line  of 
centers  and  find  its  linear  velocity  by  similar  triangles. 

31.  Slider  Crank  Mechanism.  Velocities  of  Points  in  Adjacent 
Links. — Let  Fig.  17  be  a  slider  crank  mechanism  in  which  the 


FIG.  17. 

V~K  is  desired,  having  given  the  V~J  with  the  link  A  stationary. 
The  links  to  be  considered  are  A,  B  and  C,  having  centres  ab,  ac, 
and  be,  of  which  ab  and  ac  are  stationary  and  be  is  the  common 
point. 

Lay  off  a  line  representing  the  V~J  and  by  similar  triangles 
find  the  V~bc.  Then  revolve  K  about  ac  to  the  line  of  centers 
and  find  its  linear  velocity. 

Notice  that  the  above  figure  represents  the  crank,  connecting 
rod  and  crosshead  of  an  ordinary  steam  engine;  the  frame  and 
crosshead  guides,  being  represented  by  the  link  A. 

32.  Velocities  in  Non-adjacent  Links. — In  the  slider  crank 
mechanism  of  Fig.  18  let  the  points/  and  K  be  in  the  links  B  and 
D  respectively  and  the  link  A  stationary. 

The  centres  of  the  links  A ,  B  and  D  are  ab,  ad  and  bd.  Revolve 
J  about  ab  into  the  line  of  centers  and  lay  off  a  line  to  represent 
its  linear  velocity,  then  find  the  V~bd,  the  common  point. 

The  stationary  centro  ad  is  at  infinity,  as  it  is  the  point  in 


RELATIVE  LINEAR  VELOCITIES  $3 

A  about  wrhich  D  rotates.  The  point  K  in  being  revolved  about 
ad  into  the  line  of  centers  describes  a  straight  line  instead  of  an 
arc,  or  it  is  an  arc  of  infinite  radius.  It  will  be  noticed  that  the 
V~K  is  the  same  as  the  V~bd,  the  common  point. 

The  cases  given  above  are  but  a  few  examples  of  the  many  that 
might  arise.     Remember  that  the  velocities  are  the  instantaneous 


FIG.  18. 

linear  velocities,  or  velocities  for  the  positions  of  the  links  shown. 
For  any  other  position  of  the  links  the  velocities  would  be  differ- 
ent. Even  though  the  linear  velocity  of  the  point  J  remained 
constant,  the  linear  velocity  of  the  point  K  would  change  for 
every  position  of  the  links. 

PROBLEMS 

15.  Take  the  link  D  stationary  in  Fig.  13  and  assume  the  linear  velocity 
of  J"  in  link  A,  to  find  the  linear  velocity  of  K  in  the  link  C. 

16.  Take  the  link  C  stationary  in  Fig.  16  and  find  the  V~K,  assuming  the 
V-J.    ' 

17.  Take  the  link  B  stationary  in  Fig.  17  and  J  in  the  link  A.     Assume 
the  V-J  and  find  the  V~K. 

18.  (a)  Locate  all  of  the  centres. 

(6)  With  the  link  A  stationary  and  the  link  B  making  two  revolutions 
per  second,  find  the  instantaneous  linear  velocity  of  K  when  the  angle 
1-2-7  =  60  degrees. 


24  MECHANISM 

Data:  Length  1-2  =  7|";  2-3  =  6$";  3-4  =  5£";  4-5  =  6$";  6-1  =  5*"; 
6-7  =  5i";  6-5  =  l£";  7-2  =  2". 

Angle  1-2-7  =  60  degrees. 

Note. — Make  a  full  size  drawing  using  pencil  lines  only.  Draw  circles 
^  in.  diameter  at  all  pin  joints.  In  plotting  the  velocity  of  J  use  a  scale  of 
1  in.  =  10  in.  Put  statement  of  problem  in  upper  right-hand  corner  of  sheet. 
Time,  2  hours. 


CHAPTER  IV 
VELOCITY  DIAGRAMS 

33.  Velocity  Diagrams. — It  is  often  desirable  to  find  the  velocity 

of  a  point  for  more  than  one  instant,  and  this  may  be  done  by 
means  of  velocity  triangles.  There  are  two  ways  of  representing 
the  velocity,  depending  upon  whether  the  point  in  question  has  a 
motion  of  translation  or  rotation,  the  former  being  shown  by  a 
linear  velocity  diagram  and  the  latter  by  a  polar  velocity  diagram. 

34.  Linear  Velocity  Diagram. — A  linear  velocity  diagram  is 
one  in  which  the  velocities  are  plotted  by  using  rectangular 
coordinates. 

In  the  several  positions  of  the  block  A,  shown  in  Fig.  19,  let 


FIG.  19. 

a  length  of  line  represent  the  instantaneous  velocity  of  a  point  in 
the  block,  for  each  of  these  positions  each  line  being  drawn  to  the 
same  scale.  Revolve  each  velocity  around  the  point,  perpen- 
dicular to  its  direction  of  motion,  and  through  each  of  the  points 
cut  off  on  the  perpendiculars,  draw  a  smooth  curve,  which  is  the 
velocity  diagram. 

To  find  the  linear  velocity  of  the  block  at  any  intermediate 
position,  draw  a  line  through  the  diagram  perpendicular  to  the 
direction  of  motion,  and  the  length  of  line  cut  off  between  the 
upper  and  lower  sides  of  the  shaded  portion  will  be  the  linear 
velocity  for  that  position. 

35.  Polar  Velocity  Diagram. — The  polar  velocity  diagram 
differs  from  the  linear  velocity  diagram  in  that  polar  coordinates 
are  used  instead  of  rectangular  coordinates. 

In  Fig.  20  let  A  be  a  crank  revolving  about  the  center  0,  and 
let  the  velocity  of  the  outer  end  in  its  various  positions  be  repre- 
sented by  a  length  of  line,  each  of  these  lengths  being  drawn  to 

25 


26 


MECHANISM 


the  same  scale.  About  the  point  in  its  several  positions,  revolve 
the  velocity  into  the  radius  and  draw  a  smooth  curve  through 
the  points  cut  off  on  the  radii. 

The  velocity  at  any  intermediate  point  can  then  be  assumed 
to  be  equal  to  the  length  of  radial  line  cut  off  between  the  sides 
of  the  diagram. 


FIG.  20. 

36.  Velocity  Diagram  of  Engine  Crank  and  Crosshead. — In 

Fig.  21  let  0  be  the  crank  shaft,  OA  the  crank,  AB  the  connecting 
rod,  and  B  the  cross  head  of  an  ordinary  steam  engine.  If  we 
know  the  length  and  revolutions  per  minute  of  the  crank,  its 
linear  velocity  can  be  found  from  the  equation  V~  =  2nXOA  Xn 
(Art.  14). 


FIG.  21. 

Then  if  the  length  of  the  connecting  rod  is  also  known  the 
various  instantaneous  velocities  of  the  crosshead  can  be  found. 
The  instantaneous  velocities  of  the  piston  will  be  the  same  as 
those  of  the  crosshead. 

The  angular  velocity  of  the  crank  can  be  assumed  to  be  con- 
stant throughout  the  revolution. 

Draw  the  crank  pin  circle,  Fig.  22,  and  divide  it  into  any  con- 
venient number  of  parts  as  12,  corresponding  to  different  crank 
positions.  Find  the  extreme  crosshead  positions  by  taking  0  as 


VELOCITY  DIAGRAMS 


27 


a  center  and  radii,  length  of  connecting  rod  plus  length  of  crank, 
and  length  of  connecting  rod  minus  length  of  crank,  giving  M 
and  N  respectively.  For  the  intermediate  positions  of  the 
crosshead  take  a  radius  equal  to  the  length  of  connecting  rod 
and  centers  at  1,  2,  3,  4  .  .  .  12  on  the  crank  pin  circle  and 
strike  arcs  cutting  the  center  line  of  the  crosshead  in  B,  2lf 
3j,  4t  and  5r 

Lay  off  a  length  of  line,  drawn  to  any  convenient  scale  to 
represent  the  V~  of  the  crank  pin  A,  and  revolve  it  around  into 
the  radius  to  get  a  point  on  the  polar  diagram. 


FIG.  22. 

Since  we  assumed  the  angular  velocity  of  the  crank  constant 
for  all  parts  of  the  revolution,  the  linear  velocity  of  any  point 
will  also  be  constant,  and  the  polar  diagram  becomes  a  circle,  the 
radius  of  which  is  the  length  of  the  crank  plus  the  length  of  line 
representing  the  linear  velocity  of  the  crank  pin.  Knowing  the 
direction  of  motion  of  each  end  of  the  connecting  rod,  its  instan- 
taneous center  F  can  be  found  by  drawing  lines  perpendicular  to 
the  direction  of  motion  and  their  intersection  will  give  the 
instantaneous  center  or  centro  (Art.  21). 

The  linear  velocity  of  the  crosshead  end  of  the  connecting  rod 
will  be  to  the  linear  velocity  of  the  crank  pin  end  directly  as  their 
instantaneous  radii,  or 

V-B     BF       ,  BF     „. 


28 


MECHANISM 


Instead  of  locating  the  instantaneous  center  of  the  connecting 
rod  for  each  of  its  positions,  we  may  draw  through  the  various 
points  on  the  polar  diagram,  1',  2',  3'  ...  12',  lines  parallel  to 
the  connecting  rod  in  its  several  positions,  and  the  lengths  of 
lines  cut  off  on  the  perpendiculars  drawn  through  the  crosshead 
in  its  various  positions  gives  the  linear  velocities  of  the  crosshead 
J9—  1",  2A  -  2",  etc.  This  is  true  because  a  plane  passed  through 
a  triangle  parallel  to  the  base  cuts  off  proportional  parts  from  the 
other  sides,  and  since  the  part  cut  off  on  one  side  will  be  the  V~A, 
the  part  cut  off  on  the  other  side  will  be  the  V~B. 

After  locating  the  points  1",  2",  3"  ...  6",  draw  a  smooth 
curve  through  them.  This  curve  will  be  the  linear  velocity 
diagram  for  the  cross  head.  The  curve  above  and  below  the  line 
is  symmetrical,  so  that  it  is  not  necessary  to  draw  the  lower 
portion. 

37.  In  order  to  compare  the  linear  velocity  of  the  crosshead 
with  that  of  the  crank  pin  so  that  the  difference  in  their  velocities 


10 


FIG.  23. 


can  be  seen  at  a  glance,  they  can  be  superimposed,  one  on  the 
other. 

In  Fig.  23  draw  a  circle  of  radius  equal  to  the  linear  velocity  of 
the  crank  pin,  and  divide  it  into  the  same  number  of  equal  parts 
as  the  crank  pin  circle,  in  this  case  12.  Lay  off  on  the  first 
crank  position,  01"  =  £1"  from  Fig.  22  and  on  the  second  crank 
position,  02"  =  2^",  etc.,  all  the  way  around,  then  draw  a  smooth 
curve  through  the  points  1",  2",  3"  ...  12"  giving  the  velocity 
diagram  of  the  crosshead.  The  distance  1"!'  is  the  amount  of 
linear  velocity  of  the  crank  pin  in  excess  of  that  of  the  crosshead. 
In  some  parts  of  the  revolution  the  linear  velocity  of  the  crosshead 
is  greater  than  that  of  the  crank  pin.  This  occurs  when  the 


VELOCITY  DIAGRAMS 


29 


instantaneous  radius  of  the  crosshead  is  greater  than  that  of  the 
crank  pin. 

38.  Variable  Motion  Mechanism. — If  the  horizontal  center  line 
of  the  crosshead  does  not  pass  through  the  center  of  the  crank  as 
in  Fig.  24,  the  linear  velocity  diagram  for  the  crosshead  will  not 
be  the  same  for  the  forward  and  back  stroke.  Let  0  be  the  center 
of  the  crank  OA  and  MN  the  center  line  of  the  crosshead  travel. 
The  points  M  and  N  can  be  found  by  taking  0  as  the  center  and 
radii  equal  to  the  sum  and  difference  of  the  lengths  of  connecting 


FIG.  24. 

rod  and  crank  respectively.  Draw  a  line  through  M  and  0, 
cutting  the  crank  pin  circle  at  R,  and  through  N  and  0,  cutting 
the  crank  pin  circle  at  S.  R  and  S  then,  will  be  positions  of  the 
crank  pin  when  the  crosshead  is  at  each  end  of  its  stroke. 

If  the  crank  moves  in  a  clockwise  direction  as  indicated  by  the 
arrow,  the  crosshead  will  move  from  M  to  N  while- the  crank  is 
going  through  the  angle  ROS  which  is  less  than  180°.  The 
crosshead  moves  from  N  to  M  while  the  crank  moves  through 
the  angle  SOR  which  is  more  than  180°,  so  that  if  the  linear 
velocity  of  the  crank  pin  is  constant  throughout  the  revolution, 
the  crosshead  will  have  to  move  from  M  to  N  at  a  faster  rate 
than  it  does  when  travelling  from  N  to  M,  and  this  is  shown  by 
the  linear  velocity  diagram  of  the  crosshead,  the  area  of  the 


30 


MECHANISM 


curve  above  the  horizontal  line  MN  being  greater  than  the  area 
of  the  curve  below  the  line. 

The  construction  of  the  velocity  diagram  is  similar  to  that  of 
Fig.  22. 

39.  By  using  the  same  combination  of  links  as  shown  in  Fig. 
22,  but  by  making  what  is  there  the  crank  stationary,  and  using 
the  frame  as  one  of  the  moving  links  and  also  changing  the  pro- 
portions of  the  links  another  variable  motion  mechanism  can  be 
obtained. 


FIG.  25. 

Let  A}  B,  .C  and  D  of  Fig.  25  be  the  links,  with  A  stationary. 
Assume  that  the  end  of  the  revolving  arm  B,  which  is  fastened 
to  the  sliding  block,  has  a  constant  velocity,  represented  by  the 
length  of  line  LR,  laid  off  at  right  angles  to  B.  To  find  the  linear 
velocity  of  K,  a  point  in  the  link  D,  when  the  mechanism  is  in 
the  position  shown  by  the  full  lines. 

The  velocity  of  K  can  be  found  by  the  method  of  instantaneous 
centers  as  was  done  in  a  previous  chapter,  but  a  more  simple 


VELOCITY  DIAGRAMS  31 

way  where  the  velocity  is  to  be  found  for  a  number  of  different 
positions  is  as  follows. 

The  length  of  line  LR  represents  the  velocity  of  the  sliding 
block  C,  turning  about  the  center  ab.  To  find  the  velocity  of  C 
turning  about  the  center  ad,  resolve  LR  into  two  components, 
one  LS  at  right  angles  to  D,  and  the  other  RS  parallel  to  D.  The 
length  of  line  LS  represents  the  velocity  at  which  the  block  is 
turning  about  the  center  ad,  while  RS,  is  its  sliding  velocity 
along  D.  Knowing  the  turning  velocity  of  the  sliding  block 
around  the  center  ad,  we  can  find  the  turning  velocity  of  K 
around  the  same  center  by  means  of  similar  triangles,  by  drawing 
the  line  ad-P  through  S,  and  KP  parallel  to  LS.  The  length  of 
line  KP,  then  represents  the  linear  velocity  of  K  to  the  same  scale 
that  LR  represents  the  linear  velocity  of  the  end  of  B,  since 
linear  velocities  of  points  in  the  same  link  are  directly  propor- 
tional to  their  radii.  In  the  dotted  positions  of  the  links  is 
shown  the  velocity  of  K  for  the  same  length  of  line  representing 
the  linear  velocity  of  L. 

40.  The  link  B  is  called  the  constant  radius  arm  since  its  length 
does  not  change  and  the  center  about  which  it  rotates  is  called 
the  constant  radius  arm  center.  The  link  D  is  called  the  variable 
radius  arm,  since  its  radius  from  the  sliding  block  to  the  center 
ad  is  continually  changing.  Its  center  is  called  the  variable 
radius  arm  center. 


FIG.  26. 

41.  Whitworth  Quick  Return  Motion  Mechanism. — By  attach- 
ing a  connecting  rod  to  the  link  D,  Fig.  25,  and  a  crosshead,  or 
ram  to  the  connecting  rod,  a  mechanism,  known  as  the  Whitworth 
Quick  Return  Motion  is  obtained.  It  is  shown  in  simple  diagra- 
grammatic  form  in  Fig.  26,  while  Fig.  27  shows  a  form  used  in 
machine  tools. 

The  method  of  laying  out  the  velocity  diagrams  can  best  be 
shown  by  working  out  a  problem. 


32 


MECHANISM 


Problem. — Design  a  Whitworth  Quick  Return  Motion  Mechan- 
ism for  a  shaper,  and  construct  the  velocity  diagram  of  the  ram 
for  the  maximum  stroke. 

Yelocity  ratio  forward  to  return  stroke  2:1. 

Distance  between  constant  radius  arm  center  and  variable 
radius  arm  center  2  in. 

Center  of  variable  radius  arm  center  above  center  line  of  ram 
2J  in. 

Length  of  stroke — maximum,  5J  in. 

Length  of  stroke — minimum,  4  in. 

Length  of  connecting  rod,  12  in. 


Variable  Radius  Arm 


Slotted  Crank  to 
which  Connect- 
ing Rod  is 
Fastened. 


FIG.  27. 


Layout  the  maximum  stroke  AB,  Fig.  28,  and  draw  the  hori- 
zontal line  CD  at  a  distance  above  AB  equal  to  the  height  of  the 
variable  radius  arm  center  above  the  center  line  of  the  ram. 

With  A  as  a  center  and  a  radius  equal  to  the  length  of  the  con- 
necting rod,  draw  the  arc  mn,  and  with  the  same  radius  and 
center  B,  draw  the  arc  rs. 

Locate  0,  the  center  of  the  variable  radius  arm  by  trial,  on  the 
line  CD,  by  taking  0  as  the  center  of  a  circle,  the  circumference 
of  which  will  be  tangent  to  the  arcs  mn  and  rs. 

Through  A  and  0  draw  a  line  intersecting  the  arc  mn  at  K, 
and  one  through  B  and  0  intersecting  the  arc  rs  at  K'.  Then  OK 
and  OK'  are  the  positions  of  the  variable  radius  arm  when  the 
ram  is  at  the  extreme  ends  of  its  stroke  A  and  B  respectively. 


VELOCITY  DIAGRAMS 


33 


Draw  a  line  KKf  and  through  0}  perpendicular  to  KKf  draw 
the  line  OG  of  indefinite  length.  Locate  the  constant  radius 
arm  center  G  on  this  line  at  the  given  distance  from  0,  the  vari- 
able radius  arm  center. 

Next  locate  the  positions  of  the  constant  radius  arm  when  the 
ram  is  at  each  end  of  its  stroke. 

In  this  problem  the  velocity  ratio  is  2:1  or  it  takes  twice  as  long 
for  the  forward  stroke  as  the  return  stroke,  and  since  the  constant 
radius  arm  revolves  at  a  constant  velocity,  it  will  take  240°  of 
revolution  for  the  forward  stroke  and  120°  for  the  return  stroke. 

With  G  as  a  center  lay  off  two  lines  symmetrical  with  OG, 
making  an  angle  of  120°  with  each  other.  These  lines  will  inter- 


FIG.  28. 


sect  the  lines  through  OA  and  BO  at  L  and  L'  respectively, 
giving  the  positions  of  the  constant  radius  arm  when  the  ram  is 
at  the  ends  of  its  stroke. 

With  G  as  a  center  and  radius  GL  describe  a  circle,  which  will 
be  the  path  of  the  end  of  the  constant  radius  arm. 

Divide  the  variable  radius  arm  circle  up  into  any  convenient 
number  of  equal  parts,  as  12,  symmetrical  with  the  line  OG,  to 
represent  different  positions  of  the  variable  radius  arm  and  find 
the  corresponding  positions  of  the  ram  by  taking  centers  at  1,  2,  3 
...  12,  and  a  radius  equal  to  the  length  of  the  connecting 
rod  and  striking  arcs  cutting  the  line  A B. 

Since  the  velocity  of  the  constant  radius  arm  is  constant  for 
all  parts  of  the  revolution  we  may  represent  it  by  a  line  of  any 


34  MECHANISM 

convenient  length.  Let  LJl  represent  the  velocity  of  the  end  of 
the  constant  radius  arm,  laid  off  at  right  angles  to  the  radius. 
To  find  the  V~K,  resolve  L^R  into  two  components  (Art.  39),  one 
Lfi  perpendicular  to  the  variable  radius  arm  Kfi  and  the  other 
RS,  parallel  with  it.  The  component  L±S  is  the  linear  velocity 
with  which  Lx  is  turning  about  the  center  0,  and  the  V~Kl  which 
is  turning  about  the  same  center,  is  found  by  similar  triangles 
and  is  equal  to  K^.  Revolve  P  around  to  Pr  to  get  a  point  on 
the  polar  velocity  diagram  and  through  P'  draw  a  line  parallel 
to  the  connecting  rod,  getting  the  point  9  on  the  linear  velocity 
diagram  of  the  ram. 

Repeat  the  process  for  each  of  the  other  points  by  using  the 
same  length  of  line  to  represent  the  linear  velocity  of  the  con- 
stant radius  arm,  resolve  it  into  its  two  components,  find  the 
velocity  of  the  joint  of  the  connecting  rod  and  variable  radius 
arm,  and  from  that  the  velocity  of  the  ram. 

After  completing  the  diagram  it  will  be  noticed  that  the  veloc- 
ity for  the  forward  stroke  of  the  ram  is  very  much  more  constant 
than  for  the  return  stroke. 

The  minimum  stroke  of  the  ram  is  obtained  by  moving  the  end 
of  the  connecting  rod  Klt  toward  0  until  the  ram  moves  through 
the  required  minimum  stroke.  The  radius  OKl  will  not  however, 
be  one-half  of  the  stroke  because  0  does  not  lie  on  the  line  A  B. 


FIG.  29. 


It  will  be  seen  from  the  figures  illustrating  this  mechanism  that 
the  constant  radius  arm  and  the  variable  radius  arm,  both 
describe  complete  circles. 

42.  Oscillating  Arm  Quick  Return  Motion  Mechanism. — By 
using  the  same  combination  of  links  as  in  the  Whitworth  Quick 


VELOCITY  DIAGRAMS 


35 


Return  Motion  Mechanism,  but  with  different  proportions,  a 
mechanism  known  as  the  Oscillating  Arm  Quick  Return  Motion 
can  be  obtained. 

Let  Fig.  29  represent  it  in  the  same  manner  that  Fig.  26  repre- 
sented the  Whitworth  Quick  Return  Motion.  Note  that  the 
distance  between  the  fixed  centers  is  much  greater  and  that  the 
variable  radius  arm  D  is  much  loriger  than  the  constant  radius 
arm,  so  that  while  the  constant  radius  arm  revolves  in  a  com- 
plete circle,  the  variable  radius  arm  oscillates.  The  angle  through 
through  which  it  oscillates  is  found  by  drawing  lines  from  the 
variable  radius  arm  center,  tangent  to  the  circle  described  by 
the  end  of  the  constant  radius  arm. 

Except  for  the  differences  noted,  the  method  of  laying  out 
the  velocity  diagram  of  the  ram  is  similar  to  that  of  the  Whit- 
worth  Quick  Return  Motion. 

PROBLEMS 

19.  Make  sketches  showing  the  essential  difference  between  the   Whit- 
worth    quick-return    mechanism    and    the    oscillating    arm    quick-return 
mechanism. 

20.  Is  the  velocity  diagram  of  the  crosshead,  Fig.  22,.  symmetrical  about 
a  vertical  axis?     Why? 

21.  Make  a  sketch  showing  how  the  velocity  diagram  for  the  crosshead  is 
obtained  when  its  path  of  travel  is  above  the  center  of  the  crank. 

22.  (a)  Lay  out  the  velocity  diagram  of  the  ram  for  the  oscillating  arm 
shaper. 

(6)  How  many  r.p.m.  must  H  make  in  order  that  the  maximum  cutting 
speed  be  60  ft.  per  minute?  Diameter  of  G  =  12".  Diameter  of  #  =  4". 


C-    -    77 


Data:     Length   of  oscillating  arm  C  =  20".     Length  of  link  5=9$" 
maximum  length  DE  =  4£".     Time  ratio  (maximum  stroke)  =5:3. 


36  MECHANISM 

Note. — Make  a   skeleton  pencil   drawing,    half  size.     Lay   off  the   line 

T\  Tfl 

representing  the  velocity  of  D  =  —=-   (not  half  size). 

o 

Put  no  statement  of  problem  or  data  on  sheet,  but  this  title: 
LAYOUT  OF  VELOCITY  DIAGRAMS 
FOR 

OSCILLATING  ARM  QUICK-RETURN  MECHANISM 
Time,  8  hours. 


CHAPTER  V 
PARALLEL  AND  STRAIGHT-LINE  MOTION  MECHANISMS 

43.  A  parallel  motion  mechanism  is  a  combination  of  links  so 
constructed  that  when  one  point  in  it  moves  in  any  path,  an- 
other point  will  move  in  a  parallel  path. 

Parallelogram. — The  parallelogram  is  one  of  the  most  simple 
mechanisms  for  giving  an  exact  parallel  motion.  It  consists 
of  four  links,  the  lengths  of  opposite  ones  being  equal. 

Fig.  30  is  a  parallelogram  in  which  p,  the  joint  of  the  two 
links  CD  and  AD  is  fixed. 


Take  a  point  P  on  the  link  AB  extended  and  draw  the  line 
PD,  cutting  the  link  BC  at  T,  then  in  the  triangles  PBT  and  PAD. 

D/TT  PR  P7? 

-r~  =  p-r  or  BT  =  -^-T  X  AD  =  a  constant. 

siU      JrJ\.  JL  A. 

Therefore  T  is  always  at  the  same  point  on  BC  for  a  given  position 
of  P  on  the  link  AB. 

DP     AP 
Also,  ™  =  -j-=,  which  is  constant  for  all  positions  of  the  linkage 

so  that  if  T  is  the  tracing  point  and  is  made  to  follow  any  out- 
line, the  pencil  point  P,  will  trace  a  similar  outline,  only  on  a 
larger  scale.  If  the  link  BC  is  extended  and  T  taken  on  the 
extension  as  in  Fig.  31,  P  will  fall  between  D  and  T,  and  P  will 
then  trace  an  outline  smaller  than  that  traced  by  T. 

By  moving  T  along  the  link  BC,  any  ratio  of  size  can  be  ob- 

DP 
tained  the  ratio  being     =. 

37 


38 


MECHANISM 


It  will  be  noticed  in  these  figures  that  the  fixed,  pencil  and 
tracing  points  lie  on  the  same  straight  line.  This  must  always 
be  true. 

Fig.  32  shows  the  fixed  point  0  on  an  extension  of  one  of  the 
links,  while  in  Fig.  33  it  is  between  the  ends  of  the  links.  The 


FIG.  31. 


FIG.  32. 

same  principles  hold  good  for  these  cases  as  in  the  first  two 
considered. 

OP 

In  Fig.  33  the  ratio  -^=  =  1  so  that  P  traces  an  exact  duplicate 

of  that  traced  by  T. 

These  mechanisms  are  used  for  copying  and  enlarging  maps, 
engravings,  etc. 

In  the  latter  case  the  pattern  is  usually  made  to  an  enlarged 
scale  to  eliminate  errors,  the  engraving  tool  being  an  end 


STRAIGHT  LINE  MOTIONS 


39 


milling  cutter  revolving  at  a  high  speed,  is  made  to  produce 
the  similar  pattern  reduced  or  enlarged  as  the  case  may  be.1 


FIG.  33. 


44.  Parallel  Ruler. — Fig.   34   shows  the   application   of  the 
parallelogram  to  the  parallel  ruler  for  drawing  parallel  lines. 


FIG.  34. 


45.  Roberval  Balance. — By  adding  a  fifth  link  to  the  parallelo- 
gram, a  double  parallelogram  is  obtained  in  which  an  application 
is  found  in  the  druggists'  balance  of  Fig.  35.  By  placing  the 


FIG.  35. 

support  midway  between  the  scale  pan  uprights,  the  load  will 
be  the  same  on  each  pan  when  they  are  at  the  same  level. 

46.  Drafting  Board  Parallel  Mechanisms. — There  are  many  of 
these  devices  for  guiding  the  straight-edge  with  a  parallel  motion, 
only  a  few  of  which  will  be  shown  here. 

1  For  a  description  of  an  engraving  machine  see  Machinery  of  April,  1911, 
page  602. 


40 


MECHANISM 


In  Figs.  36  and  37  the  straight-edge  is  made  so  that  it  projects 
beneath  the  board  at  the  ends,  as  well  as  being  on  top. 

In  Fig.  36  one  cord  is  fastened  to  the  straight  edge  where  it 


FIG.  36. 


projects  beneath  the  board  at  a,  carried  up  and  around  one 
pulley,  then  diagonally  down  across  the  board  around  a  second 
pulley  and  fastened  to  the  straight  edge  at  b.  Another  cord 


FIG.  37. 

beginning  at  b  is  carried  up  and  around  a  pulley  then  diagonally 
across  the  board  around  a  second  pulley  and  up  to  a.  Fig.  37  is 
somewhat  similar  except  that  there  are  two  pulleys  each  at  a 
and  b  and  the  cords  are  fastened  to  the  corners  of  the  board. 


-Two  Pulleys,  Vertical  Axis 


Pulley,  Horizontal  Axis 

FIG.  38. 


In  Fig.  38,  six  pulleys  are  required,  but  the  cord  is  continuous. 
Fig.  39  shows  diagrammatically  the "  Universal  Drafting 
Machine."  This  machine  takes  the  place  of  the  tee-square, 


STRAIGHT  LINE  MOTIONS 


41 


scales,  triangles  and  protractor.  The  mechanism  consists  of 
two  sets  of  parallelograms,  so  arranged  that  in  no  matter 
what  position  they  are  placed,  the  scales  will  be  parallel  to 
their  original  positions.  The  head  to  which  the  scales  are  con- 
nected is  movable  about  its  center,  thus  allowing  any  angle  of 
the  scales  with  the  horizontal,  the  most  common  angles  being 


FIG.  39. 

obtained  by  a  pin  dropping  into  holes,  and  the  others  by  means 
of  a  vernier  and  the  head  clamped  by  means  of  a  thumb  screw. 

47.  Straight -line  Motion  Mechanisms. — A  straight-line  motion 
mechanism  is  one  that  will  cause  some  point  in  the  mechanism 
to  move  in  a  straight  line  without  the  aid  of  guides. 

It  was  necessary  to  use  them  originally  when  it  was  desired 
to  make  some  part  move  in  a  straight  line,  as  for  example,  the 


FIG.  40. 

crosshead  of  an  engine,  because  machine  tools  were  not  far 
enough  developed  to  make  plane  surfaces. 

Most  of  the  so-called  straight-line  motions  are  only  approxi- 
mate, while  there  are  a  few  that  are  mathematically  correct. 

48.  Watt's  Straight -line  Motion. — This  is  the  best  known  and 
most  widely  used  of  all  the  straight-line  motions.  In  Fig.  40? 


42 


MECHANISM 


it  is  shown  in  its  simplest  form.  In  the  figure  the  connecting 
link  BC  is  shown  perpendicular  to  AB  and  CD  when  they  are 
parallel,  but  while  this  is  the  best  proportion,  it  is  not  necessary 
that  they  be  so.  The  tracing  point  T  is  located  on  BC,  where  a 
line  joining  A  and  D  intersects  it. 


FIG.  41. 

Fig.  41  shows  the  application  of  the  Watt  straight-line  motion 
in  combination  with  the  form  of  parallelogram  shown  in  Fig.  32, 
to  a  walking  beam  engine.  0  is  the  center  of  the  walking  beam, 
F,  a  point  on  the  frame  and  H,  the  crank  shaft  bearing.  The 
links  OC,  CD  and  DF  compose  the  Watt  motion  mechanism,  the 
point  T,  moving  in  a  straight  vertical  line,  while  the  links  OA, 
AE,  ED  and  DC  make  up  the  elements  of  the  parallelogram. 

The  point  T  is  found  as  in  Fig.  40  and  E  is  taken  on  a  line 


FIG.  42. 


In  practice  OC  and  FD  are  usually 
A  pump  rod  is  usually  attached  to 


drawn  through  0  and  T. 
taken  as  one-half  of  OA. 
link  CD  at  T  as  shown. 

49.  Scott  Russell  Straight-line  Motions. — In  this  mechanism 
shown  in  Fig.  42  there  are  two  links  A  and  B  and  the  sliding 


STRAIGHT  LINE  MOTIONS 


43 


block  C.  B  is  one-half  the  length  of  A  and  is  fastened  to  A  at 
its  middle  point.  Take  a  center  at  the  intersection  of  A  and 
B  and  a  radius  of  length  B;  strike  an  arc  which  will  pass  through 
the  points  T,  0  and  C,  so  that  the  angle  TOO  is  a  right  angle, 
and  T  will  move  in  a  straight  line.  This  is  an  exact  straight- 
line  motion,  but  it  requires  the  sliding  block  moving  between 
plane  guides. 


FIG.  43. 

Fig.  43  shows  a  modification  of  the  mechanism,  in  which  the 
link  D  is  substituted  for  the  sliding  block.  In  this  case,  the 
longer  the  arm  D,  the  more  nearly  will  T  move  in  a  straight  line. 
This  is  sometimes  known  as  the  "grass-hopper  motion." 

In  Fig.  44  is  shown  a  second  modification  of  the  mechanism 
that  will  give  an  approximate  straight  line.  In  this  case,  the 
arm  E  is  substituted  for  D  of  the  previous  figure. 


FIG.  44. 

50.  Thompson  Indicator  Motion. — Fig.  45  shows  the  application 
of  the  Scott  Russell  motion  to  a  Thompson  steam  engine  in- 
dicator.1 The  pencil  at  T,  which  traces  the  diagram  on  an 

1  For  various  other  forms  of  indicator  mechanisms,  together  with  the 
proportions  as  made  by  the  various  manufacturers,  see  "Machine  Design," 
Part  1,  by  Forrest  R.  Jones. 


*. 


44 


MECHANISM 


oscillating  drum,  is  guided  by  a  Scott  Russell  straight-line 
motion,  consisting  of  the  links  OA,  AT  and  CD.  The  length  of 
link  BE  is  determined  by  drawing  a  line  through  0  and  T,  and 
noting  the  point  E}  where  it  cuts  the  center  line  of  the  cylinder. 
If  instead  of  the  link  CD,  the  link  FE  were  used,  the  linkage 
would  then  be  the  parallelogram  shown  in  Fig.  32.  This  link 


FIG.  45. 

is  impossible  however  as  it  would  cut  the  side  of  the  steam 
cylinder  of  the  indicator. 

While  the  pencil  point  does  not  move  in  an  exact  straight 
line  TGj  the  error  is  slight. 

51.  Tchebicheff's  Approximate  Straight-line  Motion. — This 
mechanism  devised  by  Prof.  Tchebicheff  of  Petrograd,  is  shown 


E 


FIG.  46. 


FIG.  47. 


in  Fig.  46.  The  proportions  of  the  links  are  such  that 
AB  =  CD  =  5,  BC  =  2  and  AD  =  4.  The  tracing  point  T  in  the 
middle  of  BC  moves  in  an  approximate  straight  line  parallel 
to  AD. 


STRAIGHT  LINE  MOTIONS  45 

52.  Robert's  Approximate  Straight  -line  motion.  —  Fig.  47  shows 
this  mechanism  which  consists  of  two  equal  links  AB  and  DE 
and  a  triangular  shaped  link  BTD  in  which  BT=TD  =  AB}  and 

AE 

—~-.     The  tracing  point   T,  closely  follows  the  straight 


line  AE. 

53.  Peaucellier's  Exact  Straight  -line  Motion.  —  This  mechanism 
Fig.  48,  consists  of  seven  movable  links  and  one  stationary  link. 
The  proportions  are  such  that  AO  =  BO; 


FIG.  48. 

and  CD  =  DO.     Then  0,  C  and  T  will  always  lie  on  the  same 

straight  line. 

and  OB2-OE2^BT 2-TE*_ 

or  transposing  OB*-BT2  =  OE2-  TE2 

=  (OE  +  TE)(OE-TE) 

=OTXOC 
Therefore  OTxOC  is  a  constant. 

Now  suppose  that  the  linkage  is  moved  until  C  falls  at  Cf  and 
let  T'  be  the  new  position  of  T. 
Then  we  have  OC'XOT' =  OCxOT 

OC'  _  OT 
ir  OC~OTr 

And    OD  =  DC  =  DC'  so  that  0,  C  and  C'  lie  on  a  semi-circle 
which  makes  the  angle  OCC'  a  right  angle. 

In  the  triangles  COC'  and  TOT',  the  angle  TOT'  is  common, 
and  since  the  sides  are  proportional,  the  triangles  are  similar. 


46 


MECHANISM 


Thus  since  the  angle  OCC'  is  a  right  angle,  the  angle  TT'O 
is  also  a  right  angle. 

In  the  same  way  it  can  be  shown  for  any  other  position  of  the 
linkage,  that  the  line  TT'  is  perpendicular  to  OT',  and  therefore 
T  moves  in  a  straight  line. 

In  applying  this  motion  to  engines,  the  piston  rod  is  fastened 
to  T  and  this  takes  the  place  of  the  usual  crosshead  and  guides. 

If  the  link  DC  is  not  made  equal  to  DO,  T  will  describe  a 
circular  arc.1 

CD 
If  —  —  <  1  1  the  arc  described  by  T  will  be  concave  toward  0, 

CD 


while  if 


1   the  arc  will  be  convex  toward  O. 


FIG.  49. 

54.  Bricard's  Exact  Straight  -line  Motion.— If  the  linkage  of 
Fig.  49  is  made  of  the  proportions2  such  that  AC  =  DF  =  a;  CT  = 
TD  =  b;  AF  =  c;  BC  =  DE=?,  and  BE  =  *,  the  point  T  will 
describe  a  straight  line  perpendicular  to  and  bisecting  AF. 

PROBLEM 

23.  With  the  arrangement  of  the  links  of  a  parallelogram  as  shown  in 
Fig.  30,  let  DA  =  10  in. ;  DC  =  6  in. ;  CT  =  4  in.  Will  the  copy  be  larger  or 
smaller  than  the  original  and  how  much? 

1  For  the  proof  of  this  see  "Kinematics  of  Machines,"  by  R.  J.  Durley. 

2  For  these  proportions  and  the  proof  see  Prof.  Durley's  "  Kinematics  of 
Machines,"  page  94. 


CHAPTER  VI 


CAMS 

55.  A  cam  is  commonly  a  plate  or  cylinder  that  transmits 
motion  to  its  follower  by  means  of  its  edge  or  a  groove  cut  in 
its  surface. 

The  cam  plays  a  very  important  part  in  the  construction  of 
many  machines,  among  which  are  sewing  machines,  shoe  and 
printing  machinery.  Their  shapes  are  as  numerous  as  the  uses 
to  which  they  are  put,  so  that  only  some  of  the  most  common 
forms  will  be  considered. 

56.  Contact  Between  Cam  and  Follower. — Theoretically  the 
contact  between  a  cam  and  its  follower  is  a  point  or  a  line,  see 
Fig.  50.     The  disadvantage  of  these  can  readily  be  seen,  since 


FIG.  50. 

the  wear  would  be  excessive,  so  that  for  practical  purposes  a 
roll  is  generally  substituted,  as  in  Fig.  51.  In  this  case  the  roll 
turns  on  a  pin  that  is  rigidly  fastened  to  the  follower,  so  that 
while  there  is  sliding  contact  between  the  pin  and  the  roll,  there 
is  rolling  contact  between  the  roll  and  the  cam.  This  has  the 
advantage  that  nearly  all  of  the  wear  is  concentrated  on  the 
pin  which  can  be  replaced  easily. 

47 


48 


MECHANISM 


The  ordinary  diameters  of  rolls  are  from  J  in.  to  2  in. 
diameter.  Sometimes  a  flat  face  follower  as  shown  in  Fig.  52 
is  used.  This  type  of  follower  is  better  adapted  for  use  with 
light  loads  and  a  small  rise,  than  for  heavy  loads  and  a  large  rise. 

57.  Base  Circle. — The  base  circle  is  a  circle  with  its  center 
at  the  center  of  the  cam  shaft  and  a  radius  equal  to  the  shortest 
distance  to  the  theoretical  cam  curve. 

It  is  from  the  base  circle  that  the  cam  curve  is  laid  out,  and 


FIG.  51. 


FIG.  52. 


the  base  circle  should  be  of  such  a  size  that  will  insure  an  easy 
motion  to  the  follower. 

The  larger  the  base  circle,  the  easier  the  motion  will  be  for  any 
given  rise  in  a  given  angle  as  is  shown  in  Fig.  53.  Let  Obe  the 
center  of  the  cam  shaft  and  the  angle  through  which  the  rise  is  to 
take  place  be  30°.  Let  l-2  =  2-3=3-4  =  4-5  =  the  rise  that 
is  to  be 'given  the  follower  in  30°  when  using  base  circles  of  radii 
01,  02,  03,  and  04.  It  can  be  seen  that  the  line  IV  represent- 
ing the  rise  when  using  the  base  circle  of  radius  04  is  the  most 
gradual  slope  of  any  of  the  lines. 

On  the  other  hand,  the  larger,  the  base  circle,  the  larger  the 
resulting  cam  will  be,  and  the  designer  often  has  to  make  the 
cam  small  in  order  to  get  it  into  a  certain  space,  so  that  it  is 
largely  a  matter  of  judgment  as  to  how  large  to  make  the  base 
circle,  but  it  should  be  large  enough  to  leave  plenty  of  stock 
around  the  cam  shaft. 


CAMS 


49 


When  using  a  small  base  circle  and  a  large  rise  of  follower  in 
a  small  angle  of  revolution  of  cam,  a  difficulty  that  is  apt  to 
be  encountered  is  shown  in  Fig.  54.  The  theoretical  curve  is 
shown  by  the  dot  two  dash  line,  and  the  working  curve  is  found, 
if  using  a  roll  follower  by  taking  centers  of  the  roll  on  the  theo- 
retical curve  and  striking  arcs,  then  drawing  a  smooth  curve 
tangent  to  these  arcs.  This  curve  is  the  working  curve.  At 
the  top  of  the  cam  shown  in  the  figure,  the  distance  from  the 
center  of  the  roll  to  the  working  curve,  represented  by  A  is 
more  than  the  radius  of  the  roll,  so  that  the  cam  would  not  be 
raised  to  the  proper  height  by  the  difference  between  the  radius 
of  the  roll  and  the  distance  A. 

The  remedy  for  this  is  to  use  a  larger  base  circle  or  a  smaller 
roll  or  both. 


FIG.  53. 


FIG.  54. 


58.  Motions  used  for  Cam  Curves. — The  motions  most  com- 
monly used  in  the  layout  of  cams  are  uniform,  harmonic,  uniformly 
accelerated  and  uniformly  retarded. 

59.  Uniform  Motion. — As  applied  to  the  motion  of  the  cam 
follower,  uniform  motion  means  equal  rises  of  follower  in  equal 
intervals  of    time,  the  time  being  measured    by  divisions  or 
degrees  on  the  base  circle. 

The  heart-shaped  cam  of  Fig.  55  is  an  example  of  this  kind. 

In  this  cam  the  base  circle  has  a  radius  OA  and  the  follower 
is  to  be  raised  a  distance  08  in  180°  of  revolution  of  the  cam 
with  a  uniform  motion,  and  to  drop  a  distance  08  in  180°  of 
revolution  with  a  uniform  motion. 


50 


MECHANISM 


The  method  of  laying  out  the  curve  is  as  follows: 
Divide  the  semi-circumference  of  the  base  circle  in  any  number 
of  equal  parts,  in  this  case  8,  and  after  laying  out  the  rise  08 
on  any  of  the  radial  lines,  divide  it  into  the  same  number  of 
equal  parts  as  the  semi-circumference  of  the  base  circle,  and  num- 
ber the  points  on  the  rise  to  correspond  to  the  divisions  into  which 
the  base  circle  is  divided. 

The  radial  lines  dividing  the  base  circle  can  be  assumed  to  be 
the  different  positions  of  the  center  line  of  the  follower  revolving 
around  the  cam,  as  it  is  not  possible  to  revolve  the  cam  on  the 
paper.  With  a  center  0  and  radius  01  strike  an  arc  intersecting 


Or  at  1';  with  the  same  center  and  a  radius  02  intersect  02' 
at  2'.  Do  this  for  all  the  divisions  of  the  rise,  then  draw  a 
smooth  curve  through  the  points  I',  2',  3'  ...  8'  which  will 
give  the  theoretical  curve  for  one-half  of  the  cam.  Since  the 
second  half  of  the  cam  is  the  same,  if  the  second  half  of  the  base 
circle  is  divided  into  the  same  number  of  equal  parts  as  the  first 
half,  the  same  points  on  the  radial  line  can  be  used  for  finding  the 
points  on  the  theoretical  curve. 

If  the  follower  were  of  the  knife  edge  type  shown  in  Fig.  50, 
the  curve  just  found  would  also  be  the  working  curve,  but  where 
a  roll  follower  is  used,  it  is  necessary  to  find  the  working  curve 
by  taking  a  radius  equal  to  the  radius  of  the  roll,  and  with  centers 
on  the  theoretical  curve  strike  arcs,  then  draw  a  smooth  curve 
tangent  to  these  arcs  which  will  give  the  working  curve. 


CAMS 


51 


It  will  be  seen  from  this,  that  for  the  same  theoretical  curve, 
an  infinite  number  of  working  curves  can  be  obtained  by  changing 
the  diameter  of  the  roll. 

60.  Harmonic  Motion. — The  projection  on  the  diameter  of  a 
circle  of  a  point  moving  with  uniform  velocity  in  the  circum- 
ference is  said  to  have  harmonic  motion.  The  application  of 
this  motion  to  the  follower  of  a  cam  can  perhaps  best  be  seen 
by  working  out  a  simple  problem. 


Problem. — Design  a  disk  edge  cam  that  will  give  a  reciprocating 
follower  a  harmonic  rise  of  4  in.  in  180°  of  the  cam's  revolution, 
a  harmonic  drop  of  4  in.  in  the  next  90°,  and  a  "rest"  or  "dwell" 
during  the  remaining  angle.  Cam  to  revolve  counter  clockwise. 

Layout  the  base  circle  of  radius  00,  Fig.  56,  and  divide  the  180° 
angle  through  which  the  cam  is  to  revolve  during  the  harmonic 
rise  of  the  follower  into  any  convenient  number  of  equal  parts, 
as  8,  and  draw  radial  lines  through  these  points.  On  one  of 
these  radial  lines  lay  off  08"  from  the  base  circle  equal  to  4  in. 
Take  08"  as  a  diameter  and  construct  a  semi-circle.  Divide 


52  MECHANISM 

the  circumference  of  the  semi-circle  into  the  same  number 
of  equal  parts  as  the  180°  of  revolution,  thus  getting  the 
points  1",  2",  3"  .  .  .  8".  Through  the  points  1",  2",  3" 
...  8"  drop  perpendiculars  to  the  diameter  of  the  semi- 
circle to  get  the  points  a,  b,  c  .  .  .  g.  The  distances 
Oa,  ab,  be,  etc.,  will  be  the  rise  of  the  follower  for  equal  angles 
through  which  the  cam  revolves.  With  0  as  a  center  and  radii 
Oa,  Ob,  Oc  .  .  .  08",  strike  arcs  cutting  01,  02,  03,  etc.,  at  1, 
2,  3,  etc.  Draw  a  smooth  curve  through  1,  2,  3,  etc.,  which  will 
be  the  theoretical  curve. 

The  next  angle  through  which  the  cam  turns  for  the  4  in. 
harmonic  drop  is  90°.  Divide  the  90°  into  any  convenient 
number  of  equal  parts.  Since  the  drop  is  the  same  amount,  as 
tHe  rise,  if  the  90°  is  divided  into  the  same  number  of  equal 
parts  as  the  rise,  the  same  construction  for  the  harmonic  motion 
can  be  used. 

During  the  remaining  angle  the  follower  is  to  remain  at  "  rest " 
which  means  that  during  that  part  of  the  revolution  the  follower 
does  not  move  toward  or  away  from  the  center  of  the  cam,  so 
that  the  theoretical  curve  is  the  arc  of  a  circle  with  a  center  at 
0  and  a  radius  00,  which  in  this  case  is  the  radius  of  the  base 
circle.  The  working  curve  is  found  in  the  same  way  as  was  done 
in  the  previous  case. 

61.  Uniformly  Accelerated  Motion. — This  is  motion  with  a 
uniformly  increasing  velocity.  It  is  the  easiest  motion  that 
can  be  given  to  the  follower  of  a  cam. 

When  the  follower  starts  from  rest,  the  formula  S  =  %aT2  is 
applicable,  where  S  =  space  passed  over;  a  =  acceleration  (a 
constant)  and  T=  time. 

Taking  the  successive  angles  through  which  the  cam  turns  as 
the  units  of  time,  the  space  passed  over  in 

1  unit  of  time,  S  =  %a  =  k  (a  constant). 

2  units  of  time,  S  =  2a  =  4k. 

3  units  of  time,  S  =  4  Ja  =  9k. 

4  units  of  time,  S  =  8a  = 

5  units  of  time,  S  =  12% 

So  that  if  the  angle  through  which  the  cam  turns,  and  the  total 
rise  of  the  follower  during  which  it  has  uniformly  accelerated 
motion  are  known,  by  dividing  the  angle  up  into  any  number  of 
units,  we  can  solve  for  k. 


CAMS 


53 


For  example,  if  the  total  rise  is  3  in.  and  there  are  five  units  of 
time  in  the  angle,  ^  =  ^  =  0.12  in.  Lay  off  from  the  base  circle 
on  the  line  at  the  end  of  the  first  unit  of  time  0.12  in.  and  on  the 
second  line  0.12  in.X4-0.48  in.,  on  the  third  0.12  in.X9  =  1.08 
in.,  on  the  fourth  0.12  in.  X 16  =  1.92  in.,  and  on  the  fifth  line 
0.12  in.  X  25  =  3  in.  These  are  the  various  distances  that  the 
follower  has  been  moved  away  from  the  base  circle  at  the  different 
intervals  of  time. 

Instead  of  calculating  this  amount  each  time,  a  graphical 
method  is  often  used  and  is  much  more  simple. 

It  will  be  seen  that  in  the  series  1&-4&-9&-16&-25&,  etc., 
the  distance  that  the  point  has  moved  from  the  starting  place 
at  the  end  of  successive  units  of  time,  that  the  distance  moved 
during  each  successive  unit  of  time  is  in  the  series  1-3-5-7-9,  etc. 

In  order  to  apply  this  to  the  follower  of  a  cam,  a  simple  example 
will  be  used. 


FIG.  57. 


Let  AB,  Fig.  57,  be  the  distance  that  a  reciprocating  follower 
is  to  be  raised  with  a  uniformly  accelerated  motion  while  the 
cam  turns  through  the  angle  AOQ.  Divide  the  angle  AOQ  into 
any  convenient  number  of  equal  parts  as  six,  and  on  a  line 
AC  of  indefinite  length  lay  off  units  in  the  series  1-3-5-7-9-11, 
using  as  many  numbers  in  the  series  as  there  are  divisions  in 
the  angle,  in  this  case  six.  Through  the  end  division  and  through 
B  draw  a  line;  parallel  to  this  line  and  through  the  other  points 
on  AC  draw  lines  intersecting  AB  at  5',  4',  3',  2'  and  1'.  This 
gives  divisions  on  A  B  that  are  proportional  to  those  on  AC. 


54  MECHANISM 

With  0  as  a  center  and  radii  01',  02'  .  .  .  OB  strike  arcs 
intersecting  01,  02  .  .  .  06  at  1,  2,  3  .  .  .6.  Through 
these  latter  points  draw  a  smooth  curve,  which  will  be  the 
theoretical  cam  curve. 

62.  Uniformly   Retarded   Motion. — This   is   the    opposite    of 
uniformly   accelerated   motion,    and   when   applied   to   a   cam 
curve,  the  series  is  laid  off  in  the  reverse  order  as  11—9—7—5—3—1. 
Very  often  both  motions  are  combined  so  that  the  follower 
starts  slowly  and  is  uniformly  accelerated  for  half  of  the  rise  and 
uniformly  retarded  for  the  second  half,  thus  bringing  it  to  rest 
slowly.     In  using  the  two  the  angle  must  always  be  divided 
into  an  even  number  of  equal  divisions,  as  there  will  be  the  same 
number   accelerated   as   retarded.     If  there   were   eight   equal 
divisions,   the   series  would  be    1-3-5-7-7-5-3-1,  and   should 
be  laid  out  in  the  same  way  as  was  described  for  uniformly 
accelerated  motion. 

63.  Cams  with  Off -set  Followers. — In  the  cases  considered  thus 
far,  the  center  line  of  the  follower  has  been  taken  directly  over 
the  center  of  the  cam  shaft,  but  there  are  numerous  cases  where 
the  follower  is  "off-set"  and  while  the  principle  of  laying  out 
the  curve  is  the  same  there  are  some  minor  differences  which 
will  be  illustrated  by  a  problem. 

Problem. — Design  a  disk-edge  cam  that  will  raise  a  recipro- 
cating roll  follower  3  in.  with  harmonic  motion  in  90°  of  the  cam's 
revolution;  the  follower  to  remain  at  rest  for  90°,  and  to  drop 
with  uniform  motion  in  the  remaining  angle.  Center  line  of 
follower  j  in.  to  left  of  cam  shaft  center.  Cam  to  turn 
clockwise. 

Draw  a  base  circle,  Fig.  58,  of  any  convenient  radius  and 
}  in.  to  the  left  of  center  of  cam  draw  the  center  line  of  the 
follower,  AB.  With  0  as  a  center  and  a  J  in.  radius  draw  a 
circle.  Divide  the  base  circle  by  starting  at  the  point  where  the 
center  line  of  the  follower  cuts  it,  and  draw  lines  through  the 
divisions  on  the  base  circle  and  tangent  to  the  circle  of  J  in. 
radius,  which  has  already  been  drawn.  These  lines,  as  in  the 
previous  cases,  represent  different  positions  of  the  center  line 
of  the  follower,  as  if  it  were  revolved  around  instead  of  the  cam 
turning. 

Since  the  first  angle  is  90°,  the  last  division  will  be  horizontal, 
as  the  first  one  is  vertical.  Divide  the  90°  into  any  convenient 
number  of  equal  parts  as  six,  and  lay  off  on  A B  from  its  intersec- 


CAMS 


55 


tion  with  the  base  circle,  3  in.  the  rise  of  follower.  With  3  in.  as 
a  diameter  draw  the  semi-circle,  and  divide  the  semi-circumfer- 
ence into  six  equal  parts  1',  2',  3',  4',  5'  and  6j.  Project  these 
points  on  the  diameter,  giving  the  points  lt,  21;  31;  4X,  51  and 
6j.  With  0  as  a  center  and  a  radius  011;  strike  an  arc  cutting 
the  first  division  of  time  at  1,  and  so  on  for  all  of  the  points, 
always  using  the  center  0.  Since  the  follower  is  to  remain  at 
rest  during  the  next  90°,  the  theoretical  cam  curve  will  be  the 
arc  of  a  circle  for  90°,  with  0  as  a  center  and  a  radius  06^ 


1 


FIG.  58. 

The  cam  has  now  turned  through  180°  which  leaves  180°  to 
complete  the  revolution.  Divide  the  180°  into  any  convenient 
number  of  equal  parts,  as  eight,  and  as  the  drop  is  to  be  uniform, 
also  divide  the  3  in.  into  eight  equal  parts.  Then  with  0  as  a 
center,  find  points  on  the  theoretical  curve  as  before. 

64.  The  Flat  Face  Follower. — In  any  of  the  foregoing  cams, 
the  flat  face  follower  can  be  substituted  for  the  roll  follower  and 
the  method  of  laying  out  the  cam  is  exactly  the  same  up  to  the 
point  of  laying  out  the  working  curve. 


56 


MECHANISM 


Using  the  same  data  as  was  used  in  Fig.  58,  let  us  design  the 
cam  for  a  flat  face  follower,  the  face  of  the  follower  being  perpen- 
dicular to  its  center  line. 

Let  us  assume  that  the  theoretical  curve  has  already  been 
found.  To  get  the  working  curve  draw  lines  la,  26,  3c,  etc.,  Fig. 
59,  perpendicular  to  the  center  line  of  the  follower  in  its  various 
positions.  After  these  lines  are  drawn  in  they  will  form  a  series 
of  triangles,  which  are  shown  cross-hatched  in  the  figure.  Draw 
a  smooth  curve  tangent  to  the  middle  point  of  each  of  these 
triangles,  which  will  be  the  working  curve. 


FIG.  59. 

If  it  is  impossible  to  draw  the  curve  tangent  to  all  of  the  lines 
without  crossing  some  of  the  others,  the  curve  cannot  be  found, 
and  a  larger  base  circle  must  be  taken. 

The  length  of  the  flat  face  of  the  follower  is  found  by  taking 
the  sum  of  the  two  greatest  distances  from  the  theoretical  curve 
on  the  center  line  of  the  follower  out  on  the  perpendicular  lines 
to  where  the  working  curve  is  tangent.  Take  one  of  these  on 
the  rising  part  of  the  curve  and  the  other  on  the  falling  side. 


CAMS  57 

These  distances  will  be  the  lengths  of  the  follower  face  required 
on  the  left  hand  and  right  hand  side  of  the  center  line  of  the 
follower  respectively, 

65.  Involute  Cams. — Involute  cams  are  so-called  because  the 
cam  curve  is  usually  of  involute  form.  The  follower  is  generally 
raised  in  less  than  half  of  the  revolution  of  the  cam  and  is  brought 
back  to  the  starting  point  by  gravity  or  springs.  Fig.  60  shows 


FIG.  60. 

such  a  cam.  One  use  for  them  is  on  ore  crushers  in  stamp  mills. 
Several  cams  are  placed  on  the  same  shaft  and  so  arranged  that 
they  will  raise  their  respective  rods  at  different  times  during 
the  revolution  of  the  shaft.  The  cam  may  be  so  designed  that 
the  follower  is  raised  twice  for  each  revolution  of  the  shaft. 

66.  Method  of  Laying  Out  an  Involute  Cam. — An  involute  is 
fhe  curve  generated  by  a  point  in  a  taut  string  as  it  is  unwrapped 
from  a  cylinder,  or  by  a  point  on  a  straight  edge  that  is  rolled 
on  a  cylinder. 

The  circle  from  which  the  involute  is  developed  is  called  the 
base  circle. 

Divide  the  angle  (f>,  Fig.  61,  through  which  the  involute  is  to 
be  developed  into  any  number  of  preferably  equal  parts,  as  six 
and  tangent  to  the  base  circle  through  these  points  draw  the 
lines  la,  2&,  3c  .  .  .  6/.  Where  the  lengths  of  divisions  on 
the  base  circle  are  not  too  great1  the  chord  can  be  taken  as 

1  If  the  lengths  of  divisions  are  taken  equal  to  one-tenth  the  diameter 
of  the  base  circle,  they  will  be  accurate  to  about  one-thousandth  of  an 
inch  per  step. 


58 


MECHANISM 


the  length  of  the  arc.  Lay  off  the  chord  01  on  la  getting  the 
point  a.  Twice  the  chord  01  on  26,  getting  the  point  6,  etc.,  then 
draw  a  smooth  curve  through  the  points  a,  b,  c,  .  .  .  etc., 
which  will  be  the  involute  curve. 

Length  of  Arc 
The  equation  for  the  involute  is  <£  =  Radius  of  Base  circle'  the 

angle  0  being  in  circular  measure. 

In  applying  this  to  a  cam  follower,  the  amount  that  the  fol- 
lower is  raised  is  equal  to  the  length  of  the  arc. 


Problem.  —  Through  how  many  degrees  must  an  involute  cam 
turn  in  order  to  raise  its  follower  3"  if  the  diameter  of  the  base 
circle  from  which  the  involute  is  developed  is  5"? 


(in  degrees)  = 


=  68.79°. 


The  distance  between  the  center  of  the  cam  and  the  center 
line  of  the  follower  should  be  equal  to  the  radius  of  the  base 
circle,  so  that  contact  between  the  cam  and  follower  will  be  on 
the  center  line  of  the  follower.  The  force  exerted  on  the  fol- 
lower will  then  be  more  nearly  all  used  in  raising  it. 


CAMS  WITH  OSCILLATING  FOLLOWERS 

67.  Cams  with  Oscillating  Followers. — An  oscillating  follower 
is  one  in  which  points  in  it  move  in  arcs  of  circles  instead  of  in 
straight  lines.  Fig.  62  illustrates  a  roll  follower  cam  of  this 
type,  while  Fig.  63  shows  a  flat  face  or  tangent  follower  cam. 

The  method  of  laying  out  a  cam  of  this  kind  is  somewhat  more 
difficult  than  that  for  the  reciprocating  follower. 


CAMS 


59 


Let  the  angle  ABC,  Fig.  64,  be  the  angle  through  which  an 
oscillating  follower  is  to  be  raised  with  a  harmonic  motion,  while 
the  cam  turns  through  an  angle  of  120°;  the  follower  to  remain 
at  rest  for  150°  of  revolution,  and  to  drop  with  harmonic  motion 
in  the  remaining  angle.  In  this  figure  OA  is  the  radius  of  the 


FIG.  G2. 

base  circle,  A B  the  radius  of  the  follower  arm,  and  the  arc  AC 
the  path  of  the  center  line  of  the  roll  as  the  follower  is  raised. 

As  the  center  of  the  roll  moves  along  AC  with  a  harmonic 
motion  which  cannot  be  laid  out  directly  on  the  arc,  it  is  neces- 
sary to  rectify  the  arc  on  a  straight  line  and  lay  out  the  harmonic 
motion  on  that.  Lay  off  on  AD  a  length  ^46  equal  to  the  arc 
AC.  With  AQ  as  a  diameter -lay  out  a  semi-circle  and  divide  it 


FIG.  G3. 

into  any  convenient  number  of  equal  parts  as  six,  then  project 
these  points  on  the  diameter,  getting  the  points  1,  2,  3,  4,  5  and  6. 
Lay  off  these  points  on  the  arc,  so  that  A1'  =  A1,  1'2'  =  12, 
2'3'=23,  etc.  Through  the  points  I',  2',  3'  ...  C  and 
the  center  of  the  follower  B}  draw  lines  intersecting  the  line  OD 
at  V,  2,,  31;  4,,  5,  and  6X. 

1  lj  and  2l  are  not  shown  on  the  figure  on  account  of  the  smallness  of 
the  drawing,  but  they  will  be  at  the  intersection  of  the  lines  through  1'B. 
2'B  and  OD. 


f60  MECHANISM 

Divide  the  first  120°  of  the  cams  circumference  into  six  equal 
parts  to  correspond  with  the  parts  into  which  the  semi-circle 
was  divided. 

With  0  as  a  center  lay  off  01"  =  01,,  02"  =  02,,  03"  =  03, 
.  .  .  06"  =  061.  If  the  roll  had  moved  along  the  line  AD, 
instead  of  along  the  arc,  the  points  1",  2",  3"  ...  6"  would 
be  points  on  the  theoretical  cam  curve,  but  since  it  moved  on 


FIG.  64. 
the    arc    and    the   lines    01",  02"     03' 


06"  represent 


different  positions  of  the  line  OD,  it  is  necessary  to  make  a  cor- 
rection for  the  distance  that  the  center  of  the  roll  is  away  from 
the  straight  line. 

This  can  be  done  by  drawing  through  the  points  1",  2",  3" 
*  .  .  6",  lines  making  the  same  angles  with  01",  02",  03" 
.  .  .  06"  that  1,B,  2,B,  3,B  .  .  .  6^8  make  with  OD,  and 
laying  off  l//a  =  !1l/,  2"6  =  212',  3"c  =  313'  .  .  .  6"f=61C.1 

1  The  reason  for  making  the  correction  can  be  seen  if  the  student  will 
take  a  piece  of  tracing  paper  or  cloth  and  placing  it  over  the  figure,  mark 
the  center  0.  Then  trace  any  line  as  05"  and  5"e.  Revolve  05"  around 
O  until  it  coincides  with  OD,  when  the  point  e  will  fall  on  5'. 


CAMS 


61 


A  smooth  curve  drawn  through  the  points  a,  b,  c,  d,  e  and  / 
will  give  the  theoretical  cam  curve.  Taking  points  in  this  line 
and  a  radius  equal  to  the  radius  of  the  roll  the  working  curve 
can  be  found.  The  remaining  part  of  the  cam  curve  is  found 
in  a  similar  manner. 

If  a  flat  face  follower  is  to  be  used  the  method  is  the  same  up 
to  the  point  of  finding  the  working  curve,  except  that  it  will  be 
necessary  to  draw  the  lines  I" a,  2"b,  3"c  .  .  .  etc.,  much 
longer  so  that  they  will  form  triangles  as  was  done  in  Fig.  59. 

68.  There  are  a  number  of  other  methods  of  laying  out  this  cam 
which  will  give  the  same  results;  one  of  these  is  shown  in  Fig.  65. 


FIG.  65. 

This  method  requires  a  much  larger  sheet  of  paper  for  the 
layout,  ibut  is  less  laborious  if  the  facilities  are  at  hand.  Assume 
the  data  as  in  Fig.  64  and  after  locating  the  center  of  the  oscil- 
lating arm  B  take  a  radius  OB,  and  with  0  as  a  center,  strike  an 
arc.  Divide  the  angle  through  which  the  cam  is  to  turn  to  give 
the  desired  rise  to  the  follower  up  into  six  equal  parts,  and 
through  these  points  draw  tangents  to  the  base  circle,  inter- 
secting the  arc  of  radius  OB  at  B1}  B2,  B3  .  .  .  £6.  With 
these  points  as  centers  and  a  radius  A  B,  draw  arcs  from  the 
base  circle  outward. 


62 


MECHANISM 


The  points  1',  2',  3',  4',  and  5'  on  the  arc  AC  are  found  as  i.i 
Fig.  64;  with  0  as  a  center  and  radii  01',  02',  .  .  .  OC 
strike  arcs,  getting  the  points  1",  2" ' ,  3"  .  .  .  6",  which  will 
be  points  on  the  theoretical  curve. 

PROBLEMS 

24.  Lay  out   the  theoretical  and  working  curves  for  a  disk-edge  cam 
that  will  raise  a  reciprocating  roll  follower  4  in.  with  harmonic  motion  in 
180°  of  the  cam's  revolution;  the  follower  to  remain  at  rest  for  90°,  and  to 
drop  with  uniform  motion  to  the  starting  point  in  the  remaining  angle. 
Cam  to  revolve  clockwise. 

25.  Lay  out  the  theoretical  and  working  curves  for  a  disk-edge  cam  that 
will  raise  a  reciprocating  roll  follower  3  in.  with  uniform  motion  in  120° 
of  revolution;  the  follower  to  remain  at  rest  for  60°,  and  to  drop  3  in.  with 
harmonic  motion  in  the  remaining  angle.     Cam  to  revolve  counter-clock- 
wise.    Center  line  of  follower  1  in.  to  left  of  cam  shaft  center. 

26.  Show  the  theoretical  and  working  curves  for  a  disk-edge  cam  that 
will  raise  a  reciprocating  flat-face  follower  5  in.  with  uniform  motion  in 
120°  of  the  cam's  revolution;  the  follower  to  remain  at  rest  during  the  next 
60°,  and  to  drop  with  harmonic  motion  to  the  starting  point  in  the  next 
180°.     Cam  to  revolve  counter-clockwise.     Flat  face  of  follower  to  make 
an  angle  of  75°  with  center  line  of  follower. 

27.  (a)  Design  a  disk-edge  cam  to  revolve  in  a  clockwise  direction  and 

give  its   <    fl  ,  -         >  follower  a  reciprocating  motion,  uniformly  accelerated 

upward  during  the  first  portion  of  the  cam's  revolution;  a  uniformly  retarded 
motion  upward  during  the  second  portion,  and  a  uniform  drop  during  the 
remaining  portion. 

(6)  After  laying  out  the  curve  as  in  (a),  lay  it  out  for  the  first  two  portions 
of  the  revolution  according  to  the  law  of  harmonic  motion,  the  third  portion 
being  the  same  as  before. 


Data 

1 

2 

3 

4 

Diameter  of  base  circle  inches 

41 

31 

41 

3i 

Diameter  of  roll,  inches  

H 

1JL 

IA 

14 

Diameter  of  roll  pin,  inches   

JL 

JL 

JL 

A 

Width  of  follower  inches 

£ 

| 

f 

44 

Rise  of  follower,  inches  

3f 

4 

3£ 

41 

Diameter  of  cam  shaft,  inches  

11 

li 

H 

H 

Distance  of  center  line  of  follower  to 
left  of  cam-shaft  center,  inches. 
First  portion,  degrees  

1 
90 

0 
90 

| 

60 

i 

120 

Second  portion,  degrees    

90 

90 

60 

120 

Third  portion  degrees 

180 

180 

240 

120 

CAMS 


63 


Note. — Make  an  ink  drawing,  full  size,  with  center  of  cam  shaft  ?i  in. 
from  top  border  line  and  in  the  center  of  the  sheet  right  and  left. 

Use  32  divisions  for  Nos.  1  and  2  and  36  divisions  for  Nos.  3  and  4,  laying 
them  out  upon  a  12  in.  diameter  circle  to  insure  greater  accuracy. 

Do  not  ink  in  this  circle,  and  ink  radial  lines  only  about  £  in.  farther  out 
than  roll  centers. 

If  a  flat-face  follower  is  specified,  disregard  the  dimensions  relating  to 
cam  roll  and  pin,  and  make  the  length  of  follower  face  £  in.  longer  than 
length  of  contact  on  each  side  of  center  line. 

The  object  of  drawing  the  harmonic  curve  superimposed  on  the  uniformly 
accelerated  and  uniformly  retarded  curve  is  to  see  the  difference  between 
them.  Time,  6  hours. 

28.  An  involute  cam  follower  is  to  be  raised  5£  in.     Distance  between 
center  of  cam  and  center  line  of  follower  is  6  in.     Through  how  many 
degrees  must  the  cam  turn? 

29.  What  should  be  the  distance  between  the  center  of  an  involute  cam 
and  the  center  line  of  its  follower  if  the  follower  is  raised  7  in.  in  100  degrees 
of  revolution? 

POSITIVE  MOTION  CAMS 

• 

69.  Positive  Motion  Cams. — A  positive  motion  cam  is  one 
that  does  not  depend  upon  the  force  of  gravity,  springs  or  any 
other  external  means  to  bring  the  follower  back  to  its  initial 
position. 


FIG.  66. 


A  common  method  of  making  a  positive  motion  cam  where 
a  roll  follower  is  used,  is  to  have  the  roll  move  in  a  groove  in  the 
side  of  a  disk  or  plate  as  is  shown  in  Fig.  66. 


64 


MECHANISM 


70.  Cams  of  Constant  Diameter. — A  cam  of  constant  diameter 
is  one  in  which  the  distance  between  the  sides,  measured  through 
the  center  of  the  cam  shaft,  is  constant.  It  has  the  disadvantage 
that  any  desired  law  of  motion  can  be  laid  out  for  but  180°,  the 
second  180°  being  laid  out  to  correspond.  The  layout  of  a  cam 
of  this  type  is  shown  in  Fig.  67.  Lay  out  the  first  180°  of  the 


FIG.  67. 

cam  for  the  desired  law  of  motion,  getting  the  points  1, 2,  3  .  .  . 
8  on  the  theoretical  cam  curve.  The  distance  from  the  starting 
point  through  the  center  of  the  cam  shaft  to  the  point  8  (not 
shown  in  Fig.),  will  be  the  distance  between  the  centers  of  the  rolls. 
With  this  length  as  a  radius  and  centers  at  1,  2,  3  .  .  .7  strike 
arcs  intersecting  their  respective  radii  drawn  through  the  center 
of  the  cam.  These  intersections  will  give  points  on  the  theoretical 
curve  for  the  second  half  of  the  cam. 

This  type  of  cam  can  be  used  for  heavier  work  than  the  groove 
cam  already  mentioned. 

71.  Main  and  Return  Cam. — As  the  name  implies,  this  con- 
sists of  two  cams.  The  main  cam  is  laid  out  for  any  desired 
law  of  motion  for  a  complete  revolution  or  360°.  Lay  out  the 
points  1,  2,  3  .  .  .  16  on  the  theoretical  curve,  Fig.  68, 
for  any  desired  law  of  motion  and  choose  a  distance  that  the 
rolls  are  to  be  apart.  With  this  distance  as  a  radius  and  centers 
at  1,  2,  3  .  .  .16  strike  arcs  on  the  radii  diametrically 
opposite  the  points,  which  will  give  points  on  the  theoretical 
curve  of  the  second  cam.  The  working  curves  may  then  be 
found  after  choosing  rolls  of  any  convenient  diameter.  ,  l.i  ,,:  : 


CAMS 


65 


In  this  case  as  in  that  of  the  constant  diameter  cam,  the 
follower  is  generally  constructed  so  that  it  envelops  the  shaft 
and  is  guided  by  a  square  bushing.  This  cam  combination  is 
particularly  desirable  for  heavy  work  and  slow  speeds. 


FIG.  68. 


72.  Cams  of  Constant  Breadth. — A  constant  breadth  cam  is 
one  in  which  the  distance  between  parallel  fa;ces  is  constant. 
It  is  used  with  a  flat  face  follower,  and  as  in  the  case  of  the  con- 
stant diameter  cam,  can  be  laid  out  for  any  desired  law  of  motion, 


FIG.  69. 

for  180°  only.  Assume  that  the  points  1,  2,  3  .  .  .  8,  Fig.  69, 
are  points  on  the  theoretical  cam  curve.  The  distance  08  de- 
termines the  distance  between  the  parallel  sides  of  the  follower 


66 


MECHANISM 


faces.  Through  the  points  1,  2,  3  .  .  .  8  draw  lines  perpen- 
dicular to  the  radial  lines,  and  on  the  opposite  side  of  the  cam 
center  locate  the  points  7',  6',  5'  .  .  .  1'  by  using  1,  2,  3 
.  .  .  7  as  centers  and  a  radius  08.  Through  the  points  1',  2', 
3'  ...  7'  draw  lines  parallel  to  those  drawn  through  1,  2,  3, 
...  8.  To  obtain  the  working  curve  draw  a  smooth  curve 
tangent  to  the  middle  points  of  the  triangles  formed  by  these 
perpendicular  lines,  as  was  done  in  Fig.  59. 

It  will  be  noted  that  the  above  construction  is  similar  to 
that  of  the  constant  diameter  cam  up  to  the  point  of  finding 
the  working  cam  curve. 

Fig.  70  is  a  special  type  of  constant  breadth  cam  used  on 
light  mechanisms  such  as  sewing  machines. 


Lines  joining  the  three  points  of  the  cam  form  an  equilateral 
triangle.  The  sides  are  arcs  of  circles  with  centers  at  the  points 
of  the  triangle,  and  radii  equal  to  the  chordal  distances  between 
the  points. 

73.  Cylindrical  Cams. — These  cams  may  be  used  for  either 
rectilinear  or  oscillating  motion. 

The  motion  is  obtained  by  a  groove  in  the  cam  as  in  Fig.  71 
or  by  fastening  strips  on  the  surface  of  the  cylinder  as  in  Fig.  72. 
The  latter  case  is  used  on  automatic  machines  such  as  screw 
machines  for  operating  the  turrets  and  wire  feeds,  where  it  is 
desired  to  do  different  kinds  of  work.  The  same  cam  can  be 


CAMS 


67 


used  by  making  a  different  adjustment  of  the  strips,  instead  of 
using  a  new  cam  complete. 

74.  Cylindrical  cams  can  be  laid  out  in  several  different  ways, 
one  of  which  is  to  lay  out  the  cam  without  developing  the  cylin- 
drical surface,  and  another  is  to  develop  the  surface  before 
laying  out  the  curve. 


FIG.  71. 


A  problem  will  illustrate  these  two  methods. 

Problem. — Lay  out  the  theoretical  curve  for  a  cylindrical  cam 
that  will  move  a  reciprocating  follower  a  distance  of  12  in.  to 
the  right  with  a  uniform  motion  in  1J  turns  of  the  cam;  the 
follower  to  remain  at  rest  for  J  turn;  to  move  12  in.  to  the  left 


FIG.  72. 

with  a  uniformly  accelerated  and  uniformly  retarded  motion  in 
one  turn,  and  to  remain  at  rest  for  J  turn,  thus  bringing  the 
follower  back  to  the  starting  point. 

Lay  out  the  end  and  side  views  of  the  cylinder,  Fig.  73,  and 
divide  the  circle  representing  the  end  of  the  cylinder  into  any 


68 


MECHANISM 


convenient  even  number  of  equal  parts,  as  eight.  Divide  the 
12  in.  on  the  side  view  into  the  same  number  of  equal  parts  as 
there  are  equal  divisions  in  1^  revolutions  on  the  end  view, 
or  ten,  and  number  all  of  the  divisions  to  correspond.  Through 
the  points  on  the  side  view  draw  lines  across  the  cylinder,  per- 
pendicular to  its  axis.  Then  through  the  points  1,  2,  3  .  ^-*i 
8  on  the  end  view,  draw  lines^  parallel  to  the  axis  of  the  cylinder 
until  they  intersect  the  lines  drawn  through  the  points  1,  2,  3 
.  .  .  8  on  the  side  view,  thus  getting  the  points  1',  2',  3',  etc. 


FIG.  73. 

Assume  that  the  top  of  the  cylinder  moves  away  from  you 
as  you  look  at  the  side  elevation;  then  the  curve  will  pass  across 
the  front  of  the  cylinder  and  disappear  at  4',  and  will  be  on  the 
far  side  until  the  point  8  is  reached,  when  it  will  again  pass  in 
front. 

From  2'  to  A  the  follower  is  to  remain  at  rest,  so  the  curve  can 
have  no  motion  along  the  cylinder. 

During  the  next  revolution  the  follower  is  to  move  12  in.  to 
the  left  with  uniformly  accelerated  and  uniformly  retarded 
motion.  From  A  draw  a  line  AB  of  indefinite  length  and  lay 
off  on  this  line  eight  divisions  in  the  series  1-3-5-7-7-5-3-1, 
using  any  convenient  unit. 


CAMS 


69 


,  Through.  B  and  the  point  C,  12  in.  away  from  A  draw  a  line. 
Parallel  to  BC  and  through  the  points  on  AB  draw  lines  inter- 
secting AC.  Through  these  intersections  on  AC  and  perpen- 
dicular to  the  axis  of  the  cylinder  draw  lines,  and  project  across 
from  the  points  on  the  end  view  to  get  points  on  the  theoretical 
curve  as  before. 

From  this  point  the  follower  is  to  remain  at  rest  for  J  revolu- 
tion of  the  cylinder,  so  that  the  theoretical  curve  will  be  drawn, 
from  C  to  8,  perpendicular  to  the  axis. 

75.  In  order  to  lay  out  this  same  cam  on  a  developed  surface, 

lay  out  the   circumference   and  length   of  the   cylinder,  as  in 

*ig.  74,  and  divide  the  circumference  into  the  same  number  of 


K Oirqinafereace  of  Cylinder 


FIG.  74. 

equal  parts  as  there  are  divisions  in  the  end  view  of  Fig.  73,  or 
eight.  ^ 

Divide  the  length  of  the  cylinder  up,  according  to  the  law  of 
motion  desired,  and  obtain  points  on  the  theoretical  curve  as 
shown  on  the  figure. 

;  In  cams  of  this  kind  where  the  groove  crosses  itself,  the  part 
that  follows  in  the  groove,  which  in  disk-groove  cams  is  a  roll, 
but  in  cylindrical  cams  is  the  frustrum  of  a  cone,  must  be  of 
special  shape. 

h  ..It  is  generally  made  oblong  and  of  such  a  length  that  .the 
leading  end  will  be  across  the  intersecting  groove  and  well  entered 
in  the  original  groove  before  the  widest  part  of  the  follower  has 
reached  the  intersecting  groove. 


70 


MECHANISM 


In  order  to  obtain  best  results  with  a  cylindrical  cam  the 
angle  that  the  theoretical  curve  on  the  developed  cylinder 
makes  with  a  perpendicular  to  the  axis  of  the  cam  should  not 
be  more  than  30°.  If  the  angle  exceeds  this,  make  the  diameter 
of  the  cylinder  larger. 

76.  Cylindrical  Cam  with  Oscillating  Follower. — The  layout 
for  a  cam  of  this  type  is  shown  in  Fig.  75.  The  radius  of  the 
follower  is  OA  and  turns  through  the  angle  AOB.  The  other 
data  is  the  same  as  in  the  previous  case.  It  will  be  seen  that 
the  elements  into  which  the  cylinder  is  divided,  instead  of  being 
parallel  to  the  axis  as  in  the  case  of  the  reciprocating  follower, 


Circumference  of  Cylinder 

FIG.  75. 


are  arcs  of  circles,  of  radii  OA  and  with  centers  on  a  line  through 
0  perpendicular  to  the  axis  of  the  cylinder.  Other  than  this, 
there  is  no  difference  between  the  methods  of  laying  out  the 
oscillating  and  reciprocating  follower  cams. 

77.  Inverse  Cams. — An  inverse  cam  is  one  in  which  the  roll  is 
on  the  driver  and  the  groove  in  the  follower.  Fig.  76  shows  a 
cam  of  this  type. 

The  inversion  is  made  to  avoid  "dead  points"  that  would  some- 
times occur  when  using  an  ordinary  cam  combination.  It  has 
the  disadvantage  that  the  motion  can  be  laid  out  for  only  180 
degrees.  It  is  largely  used  for  light  mechanisms  such  as  sewing 
machines. 


CAMS 


71 


Fig.  77  represents  a  "Scotch  cross-head"  which  is  a  special 
case  of  an  inverse  cam  with  a  reciprocating  follower,  and  often 
used  on  fire  engines.  In  this  case  the  groove  is  straight  and 
perpendicular  to  the  center  line  of  the  follower. 


FIG.  76. 

The  method  of  laying  out  the  inverse  cam  can  be  illustrated 
by  a  simple  problem. 

Problem. — Lay  out  the  theoretical  and  working  curves  for  an 


FIG.  77. 


inverse  cam  that  will  give  its  follower  motions  as  follows:  1  in. 
raise  during  the  first  20°  of  revolution  of  driving  arm;  £  in.  raise 
during  the  next  40°;  rest  during  the  next  30°;  J  in.  drop  during 
the  next  30°;  2  in.  raise  during  the  next  40°,  and  rest  during  the 


72 


MECHANISM 


next  20°.     Driving  arm  to  turn  counter  clockwise.     Length  of 
driving  arm  6  in. 

Draw  a  circle,  Fig.  78,  of  radius  OA  =6  in.  to  represent  the  path 
of  the  center  of  the  roll.  Draw  01,  02,  03,  04,  05  and  06  to 
represent  the  different  positions  of  the  center  line  of  the  driving 
arm,  and  through  1,  2,  3,  4,  5  draw  lines  perpendicular  to  the 
horizontal  diameter  of  the  circle  of  radius  OA,  thus  getting  the 
points  I',  2',  3',  4',  5'.  If  the  groove  were  horizontal  and  straight 
as  in  Fig.  77,  when  the  driving  arm  had  passed  through  the 
angle  AOl  with  its  initial  position,  the  follower  would  have  been 


FIG.  78. 


raised  a  distance  11',  but  since  the  follower  is  to  be  raised  but  1  in. 
in  that  angle,  the  groove  must  be  curved  upward  the  difference 
between  11'  and  1  inch,  or  the  distance  I'l^  The- point  lt  can 
be  found  directly  by  laying  off  on  the  line  11',  from  1,  1  in. 

When  the  driving  arm  passes  through  the  angle  102,  the  follower 
is  to  be  raised  \  in.,  and  since  it  has  already  been  raised  1  inch, 
lay  off  1J  in.  on  22',  getting  the  point  2r  During  the  next 
angle  the  follower  remains  at  rest,  so  3±  will  be  1J  in.  down 
from  3. 

While  the  follower  is  passing  through  the  angle  304,  the  fol- 
lower drops  }  in.,  and  as  we  already  had  a  raise  of  1J  in.,  the 
net  raise  laid  off  from  4  will  be  J  in. 

A  2  in.  raise  during  the  next  40°  makes  2J  in.  to  be  laid  off 
from  5.  During  the  last  20°  the  follower  remains  at  rest,  so  2j 
in.  will  also  be  laid  off  from  6,  to  get  the  point  6t. 


CAMS 


73 


The  points  5X  and  6t  in  this  problem  fall  below  the  original 
position  of  the  groove  at  A,  but  since  the  follower  is  2|  in.  higher 
than  it  was  there,  the  groove  must  be  the  same  amount  lower 
for  the  driving  arm  is  horizontal  in  both  cases.  Draw  a  smooth 
curve  through  the  points  1,,  2lt  3±  .  .  .  6t  to  obtain  the 
theoretical  curve.  From  2t  to  3t  and  from  5X  to  6t  the  follower 
remains  at  rest  so  these  parts  of  the  curve  should  be  arcs  of 
circles  of  radii  OA,  and  a  center  on  the  center  line  of  the  follower. 

The  working  curve  is  found  by  choosing  a  diameter  of  roll 
and  with  centers  on  the  theoretical  curve  draw  circles  the  diam- 
eter of  the  roll,  then  draw  curves  tangent  to  them. 

During  the  last  180°  of  revolution  of  the  driving  arm,  the 
roll  passes  through  the  groove  in  the  opposite  direction,  and 
causes  the  follower  to  go  very  much  below  its  original  position. 
If  this  is  not  desirable,  the  ends  of  the  groove  can  be  left  open  and 
the  roll  leave  the  groove  during  the  last  180°. 

PROBLEMS 

30.  Design  an  oscillating  arm  cam  to  raise  the  sliding  block  4  in.  with 
harmonic  motion  in  180°  of  counter-clockwise  revolution  of  the  cam;  the 
block  to  remain  at  rest  for  45°,  and  to  drop  4  in.  with  a  uniform  motion  in 
the  remaining  angle. 


,//     If  Flat  Face  Follower  is 
a  *        Speci  fied  make  1  ike  this 


Section  at  a-b 


Data:    AB  =  Q%";  BC=±\"\  CE  =  8%" 
Diameter  of  base  circle  =  6" 
Diameter  of  cam  roll  =  1TV' 
Diameter  of  cam  roll  pin  =  j" 

Note. — Make  a  full  size  ink  drawing  and  leave  off  all  dimensions.  Divide 
the  180°  angle  into  12  equal  parts  and  the  135°  angle  into  10  equal  parts. 
Time,  6  hours. 

31.  Make  a  full-size  ink  drawing  of  the  belt-shifter  cam.  The  shifter 
arms  to  move  with  uniform  acceleration  and  retardation  about  their  centers 


74 


MECHANISM 


c  and  g  while  the  cam  turns  about  its  center/.  Diameter  of  cam  rolls  1  in. 
Diameter  of  shifter  arm  studs  f  in.  6  =  angle  through  which  shifter 
arms  move. 

Note. — Find  the  centers  c  and  g  on  a  line  parallel  to  the  pulley  shaft  and 
midway  between  the  outer  ends  of  the  shifter  arms.  The  angle  cfg  =  30,  and 
/  is  located  by  drawing  through  c,  downward  to  the  right,  a  line  making  an 
angle  of  1%0  with  the  horizontal,  and  a  line  through  g  upward  to  the  right 
and  also  making  an  angle  of  1 J0  with  the  horizontal.  The  point  /  is  at  the 
intersection  of  these  lines. 


Divide  cfg  into  three  equal  parts.  With/  as  a  center  and  a  radius  fc 
strike  the  arc  cd,  thus  locating  d.  Lay  off  dee  =  6.  With  c  as  a  center 
and  a  radius  cd  strike  the  arc  de.  Then  d  and  e  are  the  two  extreme  positions 
of  the  cam  roll.  The  movement  of  the  other  roll  is  the  same.  Divide  the 
angle  efk  into  six  equal  parts  and  the  arc  de  into  corresponding  parts  for 
uniformly  accelerated  and  uniformly  retarded  motion.  Time,  6  hours. 

32.  Lay  out  the  theoretical  and  working  curves  for  a  main  and  return 
cam  that  will  give  a  reciprocating  follower  a  harmonic  rise  of  6  in.  in  90° 
of  revolution,  to  remain  at  rest  for  180°,  and  a  uniform  drop  during  the 
remaining  angle.  '  Cam  shaft  revolves  clockwise. 

33.  Lay  out  the  theoretical  and  working  curves  for  a  constant-diameter 
cam  that  will  raise  a  follower  8  in.  with  harmonic  motion  in  60°;  rest  for 
60°,  and  a  uniform  rise  of  3  in.  in  the  next  60°.     Cam  revolves  clockwise. 
If  the  diameter  of  the  base  circle  is  6  in.  what  is  the  correct  distance  between 
the  rolls? 

34.  Make  a  sketch  showing  the  theoretical  curve  for  a  cylindrical  cam 
that  will  give  a  reciprocating  follower  a  harmonic  motion  of  8  in.  to  the  left 
in  one  revolution  of  the  cam;  to  remain  at  rest  for  \  revolution;  to  move 
uniformly  to  the  left  2  in.  in  \  revolution,  and  a  uniform  motion  back  to  the 
starting-point  in  the  least  number  of  revolutions.     Top  of  cam  to  turn 
away  from  you  as  you  look  at  side  elevation. 

35.  Lay  out  the  theoretical  and  working  curves  for  a  cylindrical   cam 
that  will  give  its  oscillating  follower  a  harmonic  motion  from  A  to  C  in  J 
revolution;  the  follower  to  remain  at  rest  for  £  revolution;  to  return  from 
C  to  A  in  $  revolution  with  uniform  motion,  and  remain  at  rest  for  £ 
revolution. 


Diameter  of  base  of  roll  £  in. 
Depth  of  groove  \  in. 


CAMS 


75 


Note. — Make  the  views  as  shown,  in  ink  and  full  size  Put  no  statement 
of  the  problem  on  the  sheet.  Time,  8  hours. 

36.  Make  a  sketch  showing  the  theoretical  curve  for  an  inverse  cam 
that  will  raise  its  reciprocating  follower  1  in.  during  the  first  30°  revolution 
of  the  driving  arm;  to  raise  |  in.  during  the  next  30°;  to  rest  during  the  next 
30°;  to  raise  2  in.  during  the  next  20°;  to  drop  1£  in.  during  the  next  20° 
'and  to  raise  1£  in.  during  the  next  50°.  Driving  arm  6£  in.  long  and  turns 
counter-clockwise. 


CHAPTER  VII 
GEARING  FOR  PARALLEL  SHAFTS 

78.  There  are  three  separate  cases  for  which  gears  can  be 
used. 

a.  Constant  velocity 

First.     To  connect  parallel  shafts.      •  ,    ,ra  .1O* ,          ,     ., 

0.  Variable   velocity 

ratio. 
Second.     To  connect  inter-    f  a.  Constant  velocity  ratio.     r  , 

secting  shafts.  {  b.  Variable  velocity  ratio. 

Third.     To  connect  shafts  that  are  neither  parallel  nor  inter- 
secting. 

The  first  two  cases  with  a  constant  velocity  ratio  are  the  most 
common. 

FRICTION  GEARING 

79.  Friction  gearing  may  be  said  to  be  gearing  in  which  the 
motion  or  power  that  is  transmitted  depends  upon  the  friction 
between  the  surfaces  in  contact.     In  this  case  the  power  that 
can  be  transmitted  at  a  constant  velocity  ratio  is  limited,  for  as 
soon  as  slipping  occurs,  the  velocity  ratio  changes. 

80.  Rolling  Cylinders. — This  is  the  most  simple  friction  gear 
combination  and  is  used  to  transmit  motion  between  parallel 
shafts  with   a  constant  velocity  ratio.     Fig.  79  shows  such  a 
combination. 

Velocity  Ratio. — It  has  been  stated  before  (Art.  15)  that  the 
angular  velocities  of  two  points  in  two  bodies  having  the  same 
linear  velocities,  but  different  radial  distances  from  their  centers 
of  rotation,  are  inversely  as  their  radii.  Thus  if  the  two 
cylinders  of  Fig.  79  rolling  together  without  slipping,  their 
revolutions  are  inversely  as  their  radii.  In  order  to  have  a  con- 
stant velocity  ratio,  there  must  be  pure  rolling  contact  between 

76 


GEARING  FOR  PARALLEL  SHAFTS 


77 


the  cylinders.  This  means  that  the  consecutive  points  or 
elements  on  the  surface  of  one  must  come  in  contact  with  the 
successive  points  or  elements  on  the  surface  of  the  other  in  their 
order,  or  in  other  words  there  must  be  no  slipping. 

81.  Grooved  Cylinders. — In  order  to  transmit  more  power  than 
can  be  obtained  by  smooth  cylinders,  they  are  sometimes  grooved 


FIG,  79. 

as  shown  in  Fig.  80.  With  this  arrangement  there  is  not  pure 
rolling  contact  between  the  cylinders,  except  at  a  point  about 
midway  along  the  depth  of  the  grooves. 

The  angle  included  between  the  sides  of  the  grooves  should 
not  be  more  than  30°  or  less  than  10°  in  order  to  obtain  best 
results.  If  the  angle  is  made  too  great,  the  effect  of  the  grooves 


FIG.  80. 

will  be  lost  and  the  action  will  be  more  like  the  case  of  plain 
cylinders,  while  if  the  angle  is  made  too  small,  there  will  be  a 
tendency  for  the  cylinders  to  wedge  themselves  together,  and 
it  will  require  excessive  power  to  drive  them. 

82.  Evan's  Friction  Cones. — This  combination,  shown  in  Fig. 
81,  is  for  connecting  parallel  shafts  to  obtain  a  variable  velocity 
ratio.  On  the  shafts  that  are  to  be  connected  are  placed  the 


78  MECHANISM 

f  rustrums  of  two  similar  cones,  A  and  B,  with  the  large  end  of  one 
opposite  the  small  end  of  the  other,  and  between  them  an  endless 
belt  (7,  which  is  held  in  place  by  a  belt  shipper,  or  other  similar 
device. 


Velocity  Ratio. — If  A  is  the  driver,  the  revolutions  of  B  will 
be  inversely  as  the  radii  of  the  cones  where  they  are  in  contact 
with  the  belt.  Thus  if  the  radius  of  the  large  ends  of  the  cones 
is  4  in.  and  that  of  the  small  ends  2  in.,  when  the  belt  is  at  the 
large  end  of  the  driver,  the  revolutions  of  the  driven  will  be 
twice  those  of  the  driver,  while  if  the  belt  is  at  the  small  end  of 
the  driver  and  the  large  end  of  the  driven,  the  revolutions  of  the 
driven  will  be  one-half  those  of  the  driver. 

It  can  also  be  seen  from  the  figure  that  the  direction  of 
rotation,  or  the  directional  relation  is  opposite  in  the  two  shafts. 
If  it  is  desired  that  they  turn  in  the  same  direction,  a  small 
wheel  can  be  substituted  for  the  belt.  This  wheel  should  be 
covered  with  some  soft  material,  such  as  rubber  or  leather,  to 
make  good  frictional  contact.  This  combination  is  used  on 
machine  tools,  such  as  grinders  and  speed  lathes,  for  obtaining 
variable  speeds. 

83.  Seller's  Feed  Disks. — This  is"  also  a  type  of  friction  gear 
for  connecting  parallel  shafts  to  obtain  a  variable  velocity  ratio 
and  is  shown  in  Fig.  82.  The  disks  A  and  C  are  on  the  shafts 
to  which  it  is  required  to  give  the  desired  velocity  ratio. 

Two  disks  B}  having  their  inner  surfaces  slightly  convex,  fit 
loosely  on  the  same  shaft  and  engage  the  sides  of  A  and  C  near 
their  rims,  between  the  convex  surfaces. 

On  the  shaft  with  the  disks  B,  between  the  frame  and  the 
back  of  the  disks,  are  helical  springs  which  tend  to  push  the  disks 
together  and  exert  pressure  on  the  sides  of  A  and  C. 


GEARING  FOR  PARALLEL  SHAFTS 


79 


Velocity  Ratio. — If  A  is  the  driver,  the  revolutions  of  C  can  be 
increased  by  moving  the  intermediate  disks  B,  toward  A. 

This  combination  derives  its  name  from  the  fact  that  it  was 
first  used  on  the  feed  mechanisms  of  machine  tools  built  by  the 
William  Sellers  Company  of  Philadelphia. 


FIG.  82. 

84.  Bevel  Cones. — These  cones  of  which  Fig.  83  is  an  illustra- 
tion are  used  for  transmitting  motion  between  shafts  that  inter- 
sect. The  cones  need  not  be  of  the  same  size,  and  the  shafts 
need  not  intersect  at  a  right  angle. 

Velocity  Ratio. — If  the  velocity  ratio  is  as  1 : 1,  the  diameters  of 
the  bases  of  the  cones  must  be  the  same  diameter.  In  order  to 


FIG.  83. 

lay  out  a  pair  of  cones  for  any  velocity  ratio  and  shaft  angle,  the 
method  is  as  follows: 

Assume  that  the  driver  is  to  make  5  revolutions  while  the 
driven  makes  7  and  that  the  shaft  angle  is  75°.  Let  OA, 
Fig.  84,  be  the  axis  of  the  driver  and  OB  that  of  the  driven. 
At  any  point  as  a  on  OA  draw  a  perpendicular  line  ab  and  on 
this  line  lay  off  7  units;  also  at  any  point  as  c  on  OB  draw 


80 


MECHANISM 


the  perpendicular  line  cd,  and  on  it  lay  off  5  units  of  the 
same  size  as  those  laid  off  on  ab.  Through  b  and  d  draw  lines 
parallel  to  OA  and  OB  respectively;  through  their  intersection 
at  C,  draw  the  line  CO.  This  line  will  be  an  element  of  the  two 
cones,  and  cones  of  any  diameter  may  be  drawn  as  indicated 
in  the  dotted  outlines,  by  using  CO  as  the  common  element. 


FIG.  84. 

85.  Brush  Plate  and  Wheel. — This  form  of  friction  gearing 
shown  in  Fig.  85  is  for  the  purpose  of  transmitting  variable 
velocity  ratio  between  shafts  that  intersect  at  right  angles. 
It  consists  of  a  large  disk  A  and  a  small  wheel  B,  which  can 
be  moved  across  the  face  of  A,  by  means  of  a  feather  key  in  the 
shaft  of  B.  The  small  wheel  is  usually  made  of  some  softer 
material  than  the  disk  which  is  generally  of  iron,  in  order  to 
secure  more  friction  between  the  surfaces. 

The  small  wheel  should  be  the  driver  so  that  in  case  the  load 
becomes  excessive  and  there  is  slipping,  the  small  wheel  will  be 
worn  down  an  equal  amount  all  around.  If  the  large  disk 
were  the  driver  and  the  load  became  excessive,  a  flat  spot 
would  be  worn  on  the  small  wheel,  which  would  then  have  to 
be  trued  up. 

Velocity  Ratio. — The  velocity  ratio  depends  upon  the  diameter 
of  the  small  wheel  and  the  effective  diameter  of  the  disk,  that  is, 
the  diameter  of  the  path  that  the  small  wheel  makes  on  the  disk. 


GEARING  FOR  PARALLEL  SHAFTS 


81 


The  brush  plate  is  used  on  a  number  of  forms  of  sensitive 
drills  and  is  useful  in  other  cases  for  light  loads  where  slight 
variations  in  speed  are  desired 


FIG.  85. 

TOOTHED  GEARING 

86.  In  order  to  transmit  more  power  than  can  be  obtained  with 
the  rolling  cylinders  of  Fig.  79  let  us  assume  that  projections  are 
put  on  cylinder  A,  Fig.  86,  parallel  to  the  axis,  and  corresponding 


FIG.  86. 


FIG.  87. 

grooves  in  cylinder  B.  Between  the  grooves  on  B,  put  projec- 
tions, and  corresponding  grooves  on  A.  The  cylinders  will 
then  appear  as  in  Fig.  87. 


82 


MECHANISM 


The  end  view  of  the  original  cylinders  will  be  imaginary  circles, 
and  are  called  the  pitch  circles,  and  their  point  of  tangency  is  the 
pitch  point. 

Velocity  Ratio. — The  velocity  ratio  of  a  pair  of  gears  will  be 
inversely  as  the  radii  of  the  pitch  point  of  the  pitch  circles.  In 
order  that  a  pair  of  gears  transmit  a  constant  velocity  ratio,  their 
tooth  curves  must  be  such  that  a  normal  to  the  common  tangent  of 
the  teeth  at  the  point  of  contact  will  always  pass  through  the  pitch 
point.  This  statement  is  called  the  fundamental  law  of  gearing. 
The  velocity  ratio  in  this  case  will  be  constant  for  any  fraction 
of  a  revolution,  or,  in  other  words,  the  pitch  circles  have  pure 
rolling  contact  (Art.  80). 

Two  gears  the  centers  of  which  are  at  A  and  B,  Fig.  88,  and 
pitch  point  P  have  a  pair  of  teeth  in  contact  at  T  as  shown. 


Their  tooth  curves  are  so  formed  that  a  normal  to  the  common 
tangent  MN  through  T  also  passes  through  the  pitch  point  P. 
The  same  relative  motion  between  the  tooth  curves  is  obtained 
by  holding  one  gear  stationary  and  revolving  the  other  gear  and 
the  frame,  as  by  holding  the  frame  stationary  and  revolving  the 
gears.  Suppose  that  the  gear  whose  center  is  at  B  is  held  sta- 
tionary and  the  one  with  a  center  at  A  revolved.  Then  the 
direction  of  motion  of  the  contact  point  T  of  A  is  at  right  angles 
to  PT,  or  along  the  line  MN,  and  therefore  in  instantaneous 
contact  with  the  contact  portion  of  B. 


GEARING  FOR  PARALLEL  SHAFTS 


83 


Suppose  that  the  curves  are  so  formed  that  the  common 
normal  passes  through  any  other  point  Pf  instead  of  P  and  let  A 
be  revolved  as  before  in  the  direction  indicated  by  the  arrow: 
then  A  and  B  will  break  contact,  because  the  contact  point  T  of  A 
moves  toward  M  along  the  line  MN  as  before,  which  is  not  the 
direction  of  the  contact  portion  of  B.  If  A  is  revolved  in  the 
opposite  direction,  the  contact  point  T7  of  A  will  move  toward  N, 
and  the  tooth  curves  will  interfere,  hence  the  fundamental  law  of 
gearing  as  stated  above. 


FIG.  89. 


87.  Definitions  of  Tooth  Parts. — The  addendum  circle  is  the 
circle  bounding  the  ends  of  the  teeth.  See  Fig.  89. 

The  working  depth  circle  is  the  circle  below  the  pitch  circle,  a 
distance  equal  to  that  of  the  addendum  circle  above  the  pitch 
circle. 

The  dedendum  circle  or  root  circle  is  the  circle  bounding  the 
bottoms  of  the  teeth. 

The  clearance  is  the  distance  between  the  working  depth  and 
dedendum  circles. 

The  face  of  the  tooth  is  that  part  of  the  tooth  lying  between 
the  pitch  and  addendum  circle^. 

The  flank  of  the  tooth  is  that  part  of  the  tooth  lying  between 
the  pitch  and  dedendum  circle^. 

The  thickness  of  tooth  is  its  width  measured  on  the  pitch  circle. 


84  MECHANISM 

The  width  of  space  is  the  space  between  the  teeth  measured 
on  the  pitch  circle.  • 

Backlash  is  the  difference  between  the  thickness  of  a  tooth 
and  the  space  into  which  it  meshes,  measured  on  the  pitch 
circles. 

Pitch  diameter  is  the  diameter  of  the  pitch  circle,  and  is  the 
diameter  that  is  used  in  making  all  calculations  for  the  size  of 
gears. 

Pitch  is  the  measure  of  the  size  of  the  teeth.  There  are  two 
kinds  of  pitch  in  general  use  for  calculating  gear  teeth  sizes, 
circular  and  diametral. 

Circular  pitch  is  the  distance  in  inches  between  similar  points 
of  adjacent  teeth  measured  along  the  pitch  circle.  It  is  the 
sum  of  the  thickness  of  tooth  and  width  of  space,  measured 
on  the  pitch  line. 

As  there  must  be  a  whole  number  of  teeth  on  the  circum- 
ference of  a  gear,  it  is  necessary  that  the  circumference  of  the 
pitch  circle  divided  by  the  circular  pitch  be  a  whole  number. 
Thus  it  is  seen  that  if  the  circular  pitch  is  a  whole  number,  the 
diameter  of  the  pitch  circle  will  usually  be  a  number  containing 
a  fraction. 

Let  P'  =  circular  pitch  in  inches 

Let  D  =  pitch  diameter 

Let  T  =  number  of  teeth 

7rZ>or!r=^;  P'  =  ^;  D=—  .    Thus  if  P'  = 


H  in.,  T  =  36,  the  pitch  diameter  D  =  =17.19   in.   very 


nearly. 

Diametral  pitch  is  the  number  of  teeth  on  the  gear  per  inch 
of  diameter  of  the  pitch  circle. 

Let  P  =  diametral  pitch 

Let  T  =  number  of  teeth 

Let  D  =  pitch  diameter 


In  a  gear  having  64  teeth  and  a  pitch  diameter  of  16  in.,  the 
diametral  pitch  P  =  TQ  =  4-     Meaning  that    for   every   inch   of 

diameter  of  the  pitch  circle,  there  are  4  teeth  on  the  gear. 

Diametral  pitch  is  more  generally  used  than  circular  pitch 
for  the  reason  that  the  pitch  diameter  comes  out  in  equal  frac- 


GEARING  FOR  PARALLEL  SHAFTS  85 

tions,  and  the  pitch  is  deduced  from  the  diameter  and  number 
of  teeth  without  having  to  use  the  constant  TT,  which  is  necessary 
when  using  circumferences. 

It  is  sometimes  desirable  to  convert  from  one  pitch  to  the 
other.  This  can  be  done  as  follows: 

In  a  given  gear  by  the  circular  pitch  system  T  =  -^}  while  if 
using  the  diametral  pitch,  T  =  PD. 
So  that  we  may  write  -^  =PD. 

In  the  same  gear  by  the  two  systems,  the  pitch  diameter  D 
will  be  the  same,  and  as  it  appears  in  both  sides  of  the  equation 
may  be  canceled  out. 


So  that  circular  pitch  X  diametral  pitch  =  K. 

The  angle  of  action  is  the  angle  through  which  a  gear  turns 
while  a  tooth  on  it  is  in  contact  with  its  mate  on  the  other  gear. 

The  arc  of  action  is  the  arc  subtending  the  angle  of  action.  . 

The  angle  of  approach  is  the  angle  through  which  a  gear  turns 
from  the  beginning  of  contact  of  a  pair  of  teeth  until  contact 
reaches  the  pitch  point. 

The  angle  of  recess  is  the  angle  through  which  a  gear  turns 
while  contact  in  a  pair  of  teeth  is  from  the  pitch  point  until  the 
teeth  pass  out  of  contact. 

Angle  of  Approach  +  Angle  of  Recess  =  Angle  of  Action 

There  are  two  systems  of  generating  tooth  curves  in  general 
use  that  will  give  a  constant  velocity  ratio  —  the  involute  system 
and  the  cycloidal  system. 

PROBLEMS 

37.  In  a  brush  plate  and  wheel  combination,  the  diameter  of  the  small 
wheel  which  is  the  driver  is  4  in.,  and  makes  60  r.p.m.     The   minimum 
effective  radius  of  the  brush  plate  is  1J  in.,  and  its  maximum  effective 
radius  is  15  in.     What  are  its  revolutions  in  each  case  provided  there  is  no 
slipping? 

38.  Let  P  =  diametral  pitch,  P'  =  circular  pitch,    T  =  number  of  teeth 
and  D  =  pitch  diameter. 

1.  Given  P  =  10;  Z)  =  5.      Find  T. 

2.  Given  D  =15;P'  =  2.      Find  T. 

3.  Given  T  =48;  Z>  =  4£.     Find  P1  '. 

4.  Given  P'  =  li;  T  =  30.     Find  P. 

39.  In  two  spur  gears  the  driver  turns  five  times  while  driven  turns  three 


86 


MECHANISM 


times.     Circular  pitch  =  £  in.     Number  of  teeth  in  driven  30.     Find  distance 
between  centers  and  number  of  teeth  in  driver. 

40.  Two  spur  gears  are  in  mesh.     Teeth  in  driver,  80.     Circular  pitch 
=  1|  in.     Distance  between  centers  of  shafts  =  20  in.     Find  pitch  diameter 
of,  and  number  of  teeth  in,  driven. 

41.  The  distance  between  the  centers  of  two  spur  gears  that  are  in  mesh 
is  20  in.     If  one  gear  makes  three  times  as  many  revolutions  per  minute 
as  the  other  how  many  teeth  in  each  if  the  diametral  pitch  is  4? 

42.  In  an  annular  gear  and  pinion  combination,  the  distance  between 
their  centers  is  8  in.     Teeth  in  pinion  =  36.     Diametral  pitch  =  4.     What  are 
the  diameters  of,  and  how  many  teeth  in  each  of  the  gears? 

43.  Two  spur  gears  in  mesh  have  80  and  50  teeth  respectively  and  are 
of  1^  in.  pitch.     What  is  the  proper  distance  between  their  centers? 


INVOLUTE  SYSTEM 

88.  In  the  involute  system  the  tooth  outlines  are  involute 
curves. 

The  construction  for  the  involute  was  shown  in  Fig.  61. 
As  far  as  the  generation  of  the  involute  curve  is  concerned,  it 
makes  no  difference  whether  the  band  or  string  is  unwrapped 


FIG.  90. 

from  a  stationary  cylinder,  as  was  done  in  that  case,  or  the  band 
pulled  in  a  straight  line  and  the  cylinder  allowed  to  revolve  as 
in  Fig.  90.  In  this  figure  the  cylinder  A  revolves  about  its 
center,  and  the  tracing  point  T  in  the  band  traces  the  involute 
curve  ab. 

Now  suppose  that  a  second  cylinder  Bt  is  used  Fig.  91,  and 
the  band  unwrapped  from   A  to  B.     As  it  is  unwrapped  from 


GEARING  FOR  PARALLEL  SHAFTS 


87 


A,  the  tracing  point  T  will  trace  the  involute  aT  and  at  the  same 
time  it  will  trace  the  involute  dT  as  it  is  wrapped  on  B.  The 
dotted  portions  of  the  curves  Tb  and  Tc  are  the  parts  of  the 
involutes  that  will  be  traced  as  the  cylinders  are  revolved  further. 

89.  It  is  a  property  of  the  involute  that  a  normal  to  the  curve 
is  tangent  to  the  base  circle.     The  two  curves  have  a  common 
normal  at  the  point  of  contact  T,  which  is  tangent  to  both  base 
circles.     This  line  which  is  the  locus  of  all  the  points  of  con- 
tact is  called  the  line  of  action. 

90.  Spur  Gears. — Let  A  and  B,  Fig.  92,  be  the  centers  of  two 
spur  gears  with  pitch  radii  AP  and  BP  respectively. 

The  radii  of  the  base  circles  from  which  the  involutes  are  devel- 
oped are  obtained  by  drawing  the  line  of  action  ab  and  dropping 


FIG.  91. 


perpendiculars  to  it  from  the  centers  of  the  gears.  The  lengths 
of  these  perpendiculars  are  the  radii  of  the  base  circles. 

The  angle  of  obliquity  a,  is  the  angle  that  the  line  of  action 
makes  with  a  tangent  to  the  pitch  circles  drawn  through  the 
pitch  point. 

If  the  angle  of  obliquity  is  zero  the  base  circles  will  coincide 
with  the  pitch  circles,  and  the  teeth  of  the  gears  will  act  only 
at  the  pitch  point.  The  angle  of,  obliquity  can  be  taken  any 
angle  more  than  zero,  but  in  practice  it  is  usually  taken  such 


88 


MECHANISM 


that  its  sine  is  0.25  which  corresponds  to  angle  of  approximately 

Ujv 

91.  In  the  figure,  let  the  gear  with  its  center  at  A  be  the  driver 
and  revolve  clockwise  as  shown.  There  should  be  no  contact 
between  the  teeth  of  A  and  B  until  the  point  a,  where  the  line 
of  action  is  tangent  to  the  base  circle  of  the  driver,  is  reached, 
as  this  is  the  pojnt  where  the  involute  is  first  begun  to  be  gener- 
ated. The  limit  of  contact  on  the  other  side  of  the  pitch  point 


FIG.  92. 

is  b,  where  the  line  of  action  is  tangent  to  the  base  circle  of  the 
driven  gear.  Contact  will  be  along  the  line  ab,  beginning  at  a 
and  ceasing  at  b  provided  the  teeth  of  the  driven  are  long  enough 
to  reach  the  point  a,  and  the  teeth  of  the  driver  of  sufficient 
length  to  reach  the  point  b. 

If  the  teeth  of  the  gears  are  not  long  enough  for  contact  to 
begin  at  a  and  cease  at  b}  it  will  begin  where  the  addendum 

1  Several  years  ago  there  was  brought  out  a  system  of  involute  gear  teeth 
known  as  the  "stub  tooth"  gear  in  which  the  angle  of  obliquity  is  20°. 
These  teeth  are  wider  and  shorter  than  those  of  the  standard  involute 
tooth.  For  a  discussion  of  this  form  of  tooth  see  a  bulletin  by  the  Fellows 
Gear  Planer  Company  on  "The  Stub-Tooth  Gear." 


GEARING  FOR  PARALLEL  SHAFTS  89 

circle  of  the  driven  cuts  the  line  of  action  and  cease  where  the 
addendum  circle  of  the  driver  cuts  the  line  of  action.  Contact 
between  a  pair  of  teeth  should  be  continuous  between  the  begin- 
ning and  end  of  contact  along  the  line  ab. 

Contact  between  the  teeth  of  any  pair  of  gears  that  are  in 
mesh  always  begins  between  the  driver's  flank  and  the  follower's 
face  and  ceases  between  the  driver's  face  and  the  follower's  flank. 

For  a  pair  of  gears  having  equal  arcs  of  approach  and  recess 
to  have  continuous  contact,  that  is,  not  to  have  one  pair  of 
teeth  cease  contact  before  a  second  pair  have  begun  contact, 

•I  QQO 

the  angle  PAa  —  a,  Fig.  92,  should  not  be  less  than  — ^—  where 

T  equals  the  number  of  teeth  in  the  smaller  gear. 

It  is  desirable  to  have  more  than  one  pair  of  teeth  in  contact 
at  one  time,  or  to  have  the  pitch  arc  less  than  the  arc  of  action. 
Thus  if  the  pitch  arc  is  one-half  the  arc  of  action,  two  pairs  of 
teeth  will  be  in  contact  at  one  time. 

JP  360°  180° 

The  maximum  pitch  angle  equals  2P^a  =  2a = — ^-or  a  =  — =-  • 

Taking  a.  as  15°  the  least  number  of  teeth  that  can  be  used  is 
180 

-ypr-  =  12  teeth,  which  is  taken  as  the  smallest  standard  inter- 
changeable gear.  Since  there  cannot  be  a  fractional  tooth  on  a 
gear,  if  the  number  of  teeth  does  not  come  out  a  whole  number, 
the  next  higher  whole  number  must  be  taken. 


FIG.  93. 


92.  Involute  Rack  and  Pinion. — If  one  of  the  spur  gears  of 
a  pair  is  enlarged  until  its  diameter  is  infinite,  the  combination 
will  be  that  of  a  rack  and  pinion,  shown  in  Fig.  93,  since  a  rack 
is  the  same  as  a  spur  gear  of  infinite  diameter. 

Let  AP}  Fig.  94,  be  the  radius  of  the  pitch,  circle  of  the  pinion 


90 


MECHANISM 


in  a  rack  and  pinion  combination.  The  pitch  circle  of  the  rack 
being  of  infinite  diameter  is  therefore  a  straight  line. 

Let  ab  be  the  line  of  obliquity,  then  A  a  will  be  the  radius  of 
the  base  circle  of  the  pinion. 

If  the  pinion  is  the  driver  and  turning  as  shown,  contact  can- 
not begin  before  the  point  a,  the  tangent  point  of  the  pinion's 
base  circle  with  the  line  of  obliquity,  is  reached,  thus  limiting 
the  length  of  the  rack  teeth.  Since  the  base  line  of  the  rack  is 


FIG.  94. 

tangent  to  the  line  of  obliquity  at  infinity,  the  length  of  the 
pinion  teeth  can  theoretically  be  of  infinite  length,  but  practically 
they  are  limited  by  their  becoming  pointed. 

A  point  on  the  line  ab  will  generate  the  rack  and  pinion  teeth, 
as  in  the  case  of  spur  gears  (Art.  88).  As  the  pinion  turns  in 
the  direction  of  the  arrow,  the  rack  will  move  to  the  left  with  the 
same  linear  velocity  as  a  point  on  the  pitch  circle  of  the  pinion, 
which  is  greater  than  the  linear  velocity  of  a  point  on  the  base 

AP 
circle  of  the  pinion  in  the  ratio :  -j — 

When  the  generating  point  reaches  6,  the  point  a  on  the  rack 
will  have  reached  a',  for  in  the  triangles  PAa  and  a&a',  ab  is 
perpendicular  to  Aa  and  aa'  is  perpendicular  to  AP.  Also 

AP 

so  that  the  triangles  are  similar  and  the  angle  a'ba 


aa1 
ab 


Aa 


which  is  equal  to  the  angle  AaP,  is  a  right  angle.     Therefore 


GEARING  FOR  PARALLEL  SHAFTS  91 

the  outline  of  the  rack  tooth  is  a  straight  line  perpendicular 
to  the  line  of  obliquity. 

It  will  be  noted  that  the  directional  relation  between  the  rack 
and  pinion  is  not  constant,  but  depends  upon  the  length  of  the 
rack. 

93.  Involute  Annular  Gear  and  Pinion. — An  annular  gear  is  one 
in  which  the  teeth  are  cut  internally  on  the  pitch  circle  instead 
of  externally  as  in  the  case  of  spur  gears.  Fig.  95  illustrates 
an  annular  gear  and  pinion  combination. 


FIG.  95. 

The  method  of  laying  out  the  teeth  of  the  annular  gear  is 
precisely  the  same  as  for  the  spur  gear,  except  that  the  tangent 
points  of  the  base  circles  with  the  line  of  obliquity,  lie  on  the 
same  side  of  the  pitch  point  instead  of  on  opposite  sides  as  in 
spur  gears. 

Let  AP  and  BP,  Fig.  96,  be  the  pitch  radii  of  the  pinion  and 
annular  gear  respectively,  and  let  aP  be  the  line  of  obliquity. 
Then  the  radii  of  the  base  circles  will  be  Aa  and  Bb. 

Let  the  pinion  drive  as  shown,  then  since  contact  always 
begins  between  the  drivers  flank  and  followers  face,  and  cannot 
begin  before  the  tangent  point  of  the  pinions  base  circle  with 
the  line  of  obliquity  is  reached,  the  teeth  of  the  annular  gear 
cannot  pass  inside  the  point  a.  Ba  will  then  be  the  shortest 
radius  for  the  addendum  circle  of  the  annular  gear. 

The  length  of  the  pinion  teeth,  as  in  the  case  of  the  rack  and 
pinion  combination,  are  limited  only  by  their  becoming  pointed. 

In  the  figure,  contact  begins  where  the  addendum  circle  of  the 


92 


MECHANISM 


driven  cuts  the  line  of  obliquity  and  ceases  where  the  addendum 
circle  of  the  driver  cuts  the  line  of  obliquity. 

It  will  be  noted  that  the  sides  of  the  teeth  on  the  annular  gear 
are  concave  instead  of  convex,  as  in  the  case  of  spur  gears.  The 
annular  gear  tooth  is  exactly  the  same  as  the  space  between  the 
teeth  of  a  spur  gear  of  the  same  angle  of  obliquity,  pitch  and 
number  of  teeth,  with  the  exception  of  clearance  top  and  bottom. 


94.  Interference  of  Involute  Teeth. — If  the  teeth  of  the  gears 
of  Fig,  92  are  long  enough  to  pass  outside  of  the  points  a  and  b 
there  will  be  interference  between  this  extra  length  of  the  teeth 
and  that  part  of  the  flank  of  its  mate,  lying  inside  the  base 
circle,  unless  the  flank  is  hollowed  out,  or  the  extra  length  of 
face  rounded  off.     The  latter  method  is  generally  used  as  the 
objection  to  hollowing  out  the  flank  is  that  it  weakens  the  tooth. 
This  part  of  the  tooth  lying  between  the  base  and  working  depth 
circles  is  generally  made  radial. 

95.  Effect  of  Changing  the  Distance  between  Centers  of  In- 
volute Gears. — In  involute  gears,  the  tooth  curves  being  devel- 
oped by  a  point  on  a  line  .tangent  to  the  base  circles,  if  the 
centers  of  the  gears  are  drawn  apart,  the  same  involutes  will  be 
generated  and  the  gears  will  still  transmit  a  constant  velocity 
ratio,  even  though  the  pitch  circles  are  not  tangent. 


GEARING  FOR  PARALLEL  SHAFTS 


93 


The  gears  can  be  drawn  apart  until  the  teeth  just  mate  at 
their  ends,  but  in  this  case  the  backlash  will  be  excessive. 

The  centers  may  also  be  moved  toward  each  other  so  that  the 
pitch  circles  overlap,  the  limit  being  when  the  backlash  is  reduced 
to  zero. 

This  is  a  desirable  property  of  involute  gears  as  it  allows  gears 
to  be  used  on  shafts  that  are  not  the  exact  center  distances 
apart,  as  calculated  by  the  pitch  diameters. 

The  path  of  action  being  along  a  straight  line,  the  pressure  on 
the  bearings  is  constant  and  tends  to  beep  the  centers  as  far  apart 
as  possible,  and  prevent  rattling  if  there  is  any  looseness  in  the 
bearings.  This  property  of  involute  gears  used  to  be  considered 
objectionable  but  with  gears  and  bearings  of  modern  design  the 
pressure  is  not  excessive. 

96.  Interchangeable  Involute  Gears. — In  order  that  involute 
gears  may  be  interchangeable,  that  is,  any  gear  of  a  set  mesh 
properly  with  any  other  gear,  the  pitch  and  angle  of  obliquity  of 
one  must  be  the  same  as  the  pitch  and  angle  of  obliquity  of  the 
mating  gear. 


FIG.  97. 


97.  Standard  Sizes  of  Teeth  for  Gears. — In  the  spur  gears 
thus  far  discussed,  nothing  has  been  said  about  standard  sizes  of 
tooth  parts.  Two  gears  can  be  made  to  mesh  properly  together 
even  though  sohie  of  the  parts  are  not  the  same  size.  For 
different  pitches  there  are  standard  lengths  and  thicknesses  of 
teeth  however,  and  those  adopted  by  the  Brown  and  Sharpe 
Manufacturing  Company  and  given  in  Tables  A  and  B  on  pages 
94  and  95  are  almost  universally  used  in  this  country.  Fig.  97 
shows  parts  of  the  teeth  referred  to  in  each  column  of  the  tables. 


94 


MECHANISM 


TABLE  A 

GEAR  WHEELS 
Table  of  Tooth  Parts — Diametral  Pitch  in  First  Column 


Diametral 
pitch 

P 

Circular 
pitch 

P' 

Thickness  of 
tooth  on 
pitch  line 

t 

Addendum  or 
1" 

P 

s 

Working 
depth  of 
tooth 

D" 

Depth  of 
space  below 
pitch  line 

«+/ 

Whole 
depth  of 
tooth 

D"+f 

ft 

6.2832 

3.1416 

2.0000 

4.0000 

2.3142 

4.3142 

1 

4.1888 

2.0944 

1.3333 

2.6666 

1.5428 

2.8761 

1 

3.1416 

1.5708 

1.0000 

2.0000 

1.1571 

2.1571 

U 

2.5133 

1.2566 

.8000 

1.6000 

.9257 

1.7257 

H 

2.0944 

1.0472 

.6666 

1.3333 

.7714 

1.4381 

If 

.7952 

.8976 

.5714 

1  .  1429 

.6612 

1.2326 

2 

.5708 

.7854 

.5000 

1.0000 

.5785 

1.0785 

21 

.3963 

.6981 

.4444 

.8888 

.5143 

.9587 

2* 

.2566 

.6283 

.4000 

.8000 

.4628 

.8628 

2| 

.1424 

.5712 

.3636 

.7273 

.4208 

.7844 

3 

.0472 

.5236 

.3333 

.6666 

.3857 

.7190 

3* 

.8976 

.4488 

.2857 

.5714 

.3306 

.6163 

4 

.7854 

.3927 

.2500 

.5000 

.2893 

.5393 

5 

.6283 

.3142 

.2000 

.4000 

.2314 

.4314 

6 

.5236 

.2618 

.1666 

.3333 

.1928 

.3595 

7 

.4488 

.2244 

.1429 

.2857 

.1653 

.3081 

8 

.3927 

.1963 

.1250 

.2500 

.1446 

.2696 

9 

.3491 

.1745 

.1111 

.2222 

.1286 

.2397 

10 

.3142 

.1571 

.1000 

.2000 

.1157 

.2157 

11 

.2856 

.1428 

.0909 

.1818 

.1052 

.1961 

12 

.2618 

.1309 

.0833 

.1666 

.0964 

.1798 

13 

.2417 

.1208 

.0769 

.1538 

.0890 

.1659 

14 

.2244 

.1122 

.0714 

.1429 

.0826 

.1541 

15 

.2094 

.1047 

.0666 

.1333 

.0771 

.1438 

16 

.1963 

.0982 

.0625 

.1250 

.0723 

.1348 

17 

.1848 

.0924 

.0588 

.1176 

.0681 

.1269 

18 

.1745 

.0873 

.0555 

.1111 

.0643 

.1198 

19 

.1653 

.0827 

.0526 

.1053 

.0609 

.1135 

20 

.1571 

.0785 

.0500 

.1000 

.0579 

.1079 

22 

.1428 

.0714 

.0455 

.0909 

.0526 

.0980 

24 

.1309 

.0654 

.0417 

.0833 

.0482 

.0898 

26 

.1208 

.0604 

.0385 

.0769 

.0445 

.0829 

28 

.1122 

.0561 

.0357 

.0714 

.0413 

.0770 

30 

.1047 

.0524 

.0333 

.0666 

.0386 

.0719 

GEARING  FOR  PARALLEL  SHAFTS 

TABLE  B 

GEAR  WHEELS 
Table  of  Tooth  Parts — Circular  Pitch  in  First  Column 


95 


Circular 
pitch 

P' 

Threads 
or  teeth 
per 
inch 

1" 

Dia- 
metral 
pitch 

P 

Thickness 
of  tooth 
on  pitch 
line 

t 

Addendum 
or 
1" 

s 

Working 
depth  of 
tooth 

D" 

Depth  of 
space 
below 
pitch  line 

*+/ 

Whole 
depth  of 
tooth 

D"+f 

2 

j 

1.5708 

1.0000 

.6366 

1.2732 

.7366 

1.3732 

u 

A 

1.6755 

.9375 

.5968 

1  .  1937 

.6906 

1.2874 

U 

9 

1.7952 

.8740 

.5570 

1.1141 

.6445 

1.2016 

n 

A 

1.9333 

.8150 

.5173 

1.0345 

.5985 

1.1158 

i| 

§ 

2.0944 

.7500 

.4775 

.9549 

.5525 

1.0299 

ii7s 

H 

2.1855 

.7187 

.4576 

.9151 

.5294 

.9870 

u 

ft 

2.2848 

.6875 

.4377 

.8754 

.5064 

.9441 

H 

1 

2.3562 

.6666 

.4244 

.8488 

.4910 

.9154 

ii6« 

U 

2.3936 

.6562 

.4178 

.8356 

.4834 

.9012 

11 

1 

2.5133 

.6250 

.3979 

.7958 

.4604 

.8583 

Il35 

U 

2.6456 

.5937 

.3780 

.7560 

.4374 

.8156 

1J 

f 

2.7925 

.5625 

.3581 

.7162 

.4143 

.7724 

IA 

i? 

2.9568 

.5312 

.3382 

.6764 

.3913 

.7295 

i 

i 

3.1416 

.5000 

.3183 

.6366 

.3683 

.6866 

It 

1A 

3.3510 

.4687 

.2984 

.5968 

.3453 

.6437 

1 

It 

3.5904 

.4375 

.2785    • 

.5570 

.3223 

.6007 

Ii 

Il33 

3.8666 

.4062 

.2586 

.5173 

.2993 

.5579 

1 

1J 

3.9270 

.4000 

.2546 

.5092 

.2946 

.5492 

f 

1* 

4.1888 

.3750 

.2387 

.4775 

.2762 

.5150 

ii 

1ft 

4.5696 

.3437 

.2189 

.4377 

.2532 

.4720 

1 

1£ 

4.7124 

.3333 

.2122 

.4244 

.2455 

.4577 

i 

ii 

5.0265 

.3125 

.1989 

.3779 

.2301 

.4291 

1 

ij 

5.2360 

.3000 

.1910 

.3820 

.2210 

.4120 

$ 

U 

5.4978 

.2857 

.1819 

.3638 

.2105 

.3923 

» 

i? 

5.5851 

.2812 

.1790 

.3581 

.2071 

.3862 

| 

2 

6.2632 

.2500 

.1592 

.3183 

.1842 

.3433 

$ 

21 

7.0685 

.2222 

.1415 

.2830 

.1637 

.3052 

I7* 

2? 

7.1808 

.2187 

.1393 

.2785 

.1611 

.3003 

| 

2* 

7.3304 

.2143 

.1364 

.2728 

.1578 

.2942 

§ 

2* 

7.8540 

.2000 

.1273 

.2546 

.1473 

.2746 

1 

2§ 

8.3776 

.1875 

.1194 

.2387 

.1381 

.2575 

T\ 

2| 

8.6394 

.1818 

.1158 

.2316 

.1340 

.2498 

| 

3 

9.4248 

.1666 

.1061 

.2122 

.1228 

.2289 

A 

3i 

10.0531 

.1562 

.0995 

.1989 

.1151 

.2146 

10 

3* 

10.4719 

.1500 

.0955 

.1910 

.1105 

.2060 

* 

3i 

10.9956 

.1429 

.0909 

.1819 

.1052 

.1962 

96 


MECHANISM 


The  sizes  of  the  parts  apply  to  other  systems  of  teeth  as  well  as 
the  involute  system. 

98.  To  Lay  Out  a  Pair  of  Standard  Involute  Spur  Gears. 

Problem. — Lay  out  the  tooth  curves  for  a  pair  of  involute  spur 
gears  of  24  teeth  and  16  teeth  respectively;  2  diametral  pitch 
and  15°  angle  of  obliquity. 

24  teeth,  2  diametral  pitch  =  12  in.  pitch  diameter  for  large  gear. 
16  teeth,  2  diametral  pitch  =8  in.  pitch  diameter  for  small  gear. 
Draw  the  pitch  circles  Fig.  98  with  radii  AP  and  BP  equal  to 


FIG.  98. 

4  in.  and  6  in.  respectively,  and  through  P  draw  the  line  of 
obliquity  ab  having  the  given  angle  of  obliquity  with  the  com- 
mon tangent  to  the  pitch  circles  through  the  point  P.  Drop 
perpendiculars  from  the  centers  A  and  B,  cutting  the  line  of 
obliquity  in  a  and  b  respectively.  Then  Aa  and  Bb  will  be  the 
radii  of  the  base  circles  which  can  now  be  drawn. 

From  Table  A  on  page  94  find  the  amount  that  the  adden- 
dum, dedendum  and  working  depth  circles  differ  from  the  pitch 
circle,  and  draw  them  in. 

Divide  the  pitch  circle  of  the  smaller  gear  into  16  equal  parts 
and  that  of  the  larger  into  24  equal  parts,  which  will  give  the 


GEARING  FOR  PARALLEL  SHAFTS  97 

circular  pitch.  When  no  backlash  is  allowed,  the  thickness  of 
the  tooth,  and  the  width  of  space  measured  on  the  pitch  circle 
will  be  the  same.  Bisect  the  circular  pitch  on  each  of  the  gears 
which  will  give  32  equal  divisions  on  the  pitch  circle  of  the 
small  gear  and  48  on  the  larger  one. 

At  any  point  on  the  base  circle  of  each  gear  develop  an  invo- 
lute, and  draw  in  the  curves  between  the  base  and  addendum 
circles  through  alternate  points  on  the  pitch  circles.  This 
will  give  one  side  of  all  the  teeth  on  each  gear.  The  curve  for 
the  other  side  of  the  teeth  is  the  reverse  of  the  one  just  drawn.1 
Draw  the  part  of  the  teeth  lying  between  the  base  and  working 
depth  circles  radial,  and  put  in  a  small  arc  or  fillet  between  the 
working  depth  and  dedendum  circles.  This  fillet  should  be  as 
large  as  possible  without  passing  outside  of  the  working  depth 
circle. 

PROBLEMS 

44.  Make  a  sketch  of  a  pair  of  involute  spur  gears,  assuming  the  angle 
of  obliquity  and  diameters  of  the  pitch  and  addendum  circles.     Show  the 
path  of  contact  and  skeleton  teeth  where  contact  begins  and  ceases  only. 

45.  Make  a  sketch  of  two  involute  spur  gears  of  unequal  diameter,  with 
the  smaller  gear  driving  clockwise.     Assume  the  angle  of  obliquity  and  the 
addendum  circles  so  that  there  will  be  no  angle  of  approach,  but  a  maximum 
angle  of  recess. 

46.  Make  a  sketch  showing  the  path  of  contact  in  a  rack  and  pinion 
combination,  involute  system,  with  the  rack  driving  to  the  left.     Assume 
the  angle  of  obliquity  and  addendum  lines. 

47.  Make  a  sketch  showing  the  path  of  contact  in  an  annular  gear  and 
pinion   combination,   involute  system,   with  the  pinion   driving   counter 
clockwise.     Assume  the  angle  of  obliquity  and  the  diameters  of  addendum 
circles. 

48.  Draw  the  correct  tooth  curves  for  two  spur  gears  that  are  in  mesh. 
Involute  system.     Cut  templets  to  lay  out  the  tooth   curves  and  check 
them  for  contact.     Show  the  teeth  going  into  and  out  of  action. 

1  A  convenient  method  of  transferring  the  original  involute  to  draw  in 
the  tooth  curves  is  by  means  of  a  templet.  This  can  be  made  of  a  thin  piece 
of  soft  wood.  Put  a  fine  needle  through  the  wood,  near  one  end,  and  the 
center  of  the  gear,  then  cut  away  the  wood  to  conform  to  the  developed 
tooth  outline.  Then  swinging  the  templet  about  the  center  draw  in  one 
side  of  all  the  teeth.  Draw  the  other  side  of  the  teeth  by  turning  the 
templet  over  and  keeping  the  same  centers.  The  same  templet  can  be 
used  for  two  gears  only  when  they  have  the  same  number  of  teeth  and 
angle  of  obliquity. 


98  MECHANISM 

Data:     Number  of  teeth  in  gears  36  and  21. 

Diametral  pitch  1£;  angle  of  obliquity  15°. 
Small  gear  is  driver  and  revolves  counter-clockwise. 
Thickness  of  rim  below  tooth  bottoms  2  in. 

Note. — Make  center  line  of  gears  horizontal  and  in  center  of  sheet  with 
smaller  gear  to  the  right. 

Lay  out  the  pitch,  addendum,  dedendum,  base  and  working  depth  circles, 
and  develop  the  involute  between  the  base  and  addendum  circles.  The 
part  of  the  tooth  between  the  base  and  working  depth  circles  is  to  be  drawn 
radial.  Cut  templets  to  fit  the  tooth  outline  between  the  working  depth  and 
addendum  circles.  Space  out  the  teeth  on  the  pitch  circles.  Ink  drawing, 
slant  lettering.  Time,  5  hours. 

49.  Draw  the  correct  tooth  curves  for  an  involute  rack  and  pinion  that 
are  in  mesh.     Check  the  templet  for  contact  and  show  the  teeth  going  into 
and  out  of  action. 

Data:  Teeth  in  pinion  21.  Diametral  pitch  1  £.  Angle  of  obliquity  15°. 
Pinion  is  driver  and  turns  clockwise. 

Note. — Make  rack  horizontal  and  center  of  pinion  in  center  of  sheet  right 
and  left  and  l^in.  from  top  border  line.  Use  templet  of  21  tooth  gear  of 
problem  48.  No  statement  of  problem  to  go  on  sheet,  but  under  the  drawing 
put  INVOLUTE  RACK  &  PINION  in  vertical  capital  letters.  Ink  drawing. 
Time,  3  hours. 

50.  Draw  the  correct  tooth  curves  for  an  involute  annular  gear  and 
pinion  that  are  in  mesh. 

Data:     Same  as  problem  48.     Use  same  templets. 

Note. — No  statement  of  problem  or  data.  Only  INVOLUTE  ANNULAR 
GEAR  &  PINION.  Ink  drawing.  Time,  3  hours. 


CYCLOIDAL  SYSTEM 

99.  The  cycloidal  system,  as  the  name  implies  has  tooth  curves 
of  cycloidal  form.     It  is  an  older  system  than  the  involute,  but  is 
being  replaced  by  it  for  many  classes  of  work. 

Before  taking  up  the  tooth  outlines,  a  brief  discussion  of  the 
construction  of  the  different  cycloidal  curves  will  be  given. 

100.  Cycloid. — A  cycloid  is  the  curve  generated  by  a  point  on 
the  circumference  of  a  circle  rolling  on  a  straight  line.     Let  the 
circle  of  radius  OT,  Fig.  99,  be  the  rolling  or  generating  circle, 
which  rolls  on  the  straight  line  T6'. 

Divide  the  semi-circumference  of  the  circle  up  into  any  conve- 
nient number  of  equal  parts  as  six,  and  lay  off  on  the  straight 
line  7T  =  !T1,  1'2'  =  12,  2'3'  =  23,  3'4'=34,  4'5'=45  and5'6'  =  56. 

Through  1',  2',  3',  4',  5'  and  6'  draw  perpendiculars  to  T&, 
and  through  the  points  1,  2,  3,  4  and  5  on  the  semi-circle  draw 
lines  parallel  to  T6',  intersecting  the  diameter  of  the  semi-circle 


GEARING  FOR  PARALLEL  SHAFTS 


99 


in  lj,  2j,  0,  4t,  and  51;  and  the  perpendiculars  in  1",  2",  3",  4", 
5"  and  6". 

On  lj  1"  from  1",  lay  off  l//a  =  H1;  on  212//  from  2"  lay  off 
2"6  =  221,  etc.  Through  the  points  a,  6,  c,  d,  e,  and  6"  draw  a 
smooth  curve,  which  will  be  the  cycloid  traced  by  T  as  the  circle 
rolls  on  the  straight  line. 


FIG.  100. 


101.  Epicycloid. — The  epicycloid  is  the  curve  traced  by  a 
point  on  the  circumference  of  a  circle  as  it  rolls  on  the  outsideof 
a  second  circle,  called  the  directing  circle.  Let  OT,  Fig.  100, 


100 


MECHANISM 


be  the  radius  of  the  rolling  circle  and  AT,  the  radius  of  the 
directing  circle. 

Divide  the  semi-circumference  of  the  rolling  circle  into  any 
convenient  number  of  equal  parts  as  six,  and  lay  off  from  T, 
divisions  of  the  same  length  on  the  directing  circle.  Through 
the  points  I/,  2',  3',  ...  6'  just  found  and  through  the  center 
A  draw  lines. 

With  A  as  a  center  and  radii  Al,  A2,  A3,  .  .  .  AS  draw 
arcs  intersecting  the  diameter  of  the  rolling  circle  in  lt,  2l}  3t, 
.  .  .  5t,  and  the  radial  lines  in  1",  2",  3"  .  .  .  6".  On  l"llf 
from  1"  lay  off  l//a  =  H1;  on  2"24,  layoff  2"  b  =  22l;  on  3"3t, 
lay  off  3"c  =  33X  etc. 


FIG.  101. 

Though  the  points  a,  b,  c,  d,  e  and  6"  draw  a  smooth  curve 
which  will  be  the  cycloidal  curve. 

If  the  rolling  circle  is  larger  than  the  directing  circle  and  rolled 
internally  on  it,  an  epicycloid  will  still  be  generated. 

In  Fig.  101  let  AT  be  the  radius  of  the  directing  circle  and 
O'T  and  OT  the  radii  of  the  rolling  circles  which  roll  externally 
and  internally  on  the  directing  circle  respectively.  In  this  case, 
the  diameter  of  the  large  rolling  circle  is  taken  equal  to  the  sum 


GEARING  FOR  PARALLEL  SHAFTS 


101 


of  the  diameters  of  the  directing  circle  and  small  rolling  circle. 
The  curve  generated  by  the  large  rolling  circle  rolling  internally 
will  then  be  the  same  as  that  generated  by  the  small 'rolling 
circle  rolling  externally.  This  is  called  the  double  generation  of 
the  epicycloid,  and  will  be  referred  to  later  under  annular  gears. 

102.  Hypocycloid. — The  hypocycloid  is  a  curve  generated  by 
a  point  on  the  circumference  of  a  circle  rolling  internally  on  the 
directing  circle. 

The  construction  is  shown  in  Fig.  102,  and  is  the  sa'me  as  that 
for  the  epicycloid,  so  that  no  further  explanation  is  necessary. 


FIG.  102. 

If  the  rolling  circle  has  a  diameter  one-half  that  of  the  directing 
circle,  the  hypocyloid  formed  will  be  a  radial  line  of  the  directing 
circle. 

103.  Application  of  Cycloidal  Curves  to  Gear  Teeth. — In  Fig. 
103  let  AP  be  the  radius  of  the  pitch  circle  of  a  gear,  and  DP  the 
radius  of  a  circle  that  rolls  on  the  outside  of  the  pitch  circle  and 
traces  the  epicycloid  Pm.     CP  is  the  radius  of  a  circle  that  rolls 
on  the  inside  of  the  pitch  circle  and  traces  the  hypocycloid  Pn. 
Pm  will  be  the  face  and  Pn  the  flank  of  a  gear  tooth  which  is 
shown  shaded. 

104.  Cycloidal  Spur  Gears.— Let  AP  and  BP,  Fig.  104,  be  the 
pitch  radii  and  CP  and  DP  the  radii  of  the  rolling  circles  of  two 
cycloidal  spur  gears  that  are  in  mesh. 


102 


MECHANISM 


FIG.  103. 


GEARING  FOR  PARALLEL  SHAFTS  103 

The  rolling  circle  with  its  center  at  C,  when  rolled  on  the 
inside  of  the  lower  pitch  circle  generates  the  hypocycloid  Pn, 
and  when  rolled  on  the  outside  of  the  upper  pitch  circle  generates 
the  epicycloid  Pr.  These  curves  are  the  flank  of  the  lower  gear 
and  the  face  of  the  upper  gear  respectively,  and  are  generated 
simultaneously. 

The  rolling  circle  with  a  center  at  D  when  rolled  on  the  inside 
of  the  upper  pitch  circle  generates  the  hypocycloid  Pm,  and 
when  rolled  on  the  outside  of  the  lower  pitch  circle  generates  the 
epicycloid  Ps.  These  curves  are  respectively  the  flank  and  face 
of  the  teeth  of  B  and  A,  and  are  also  generated  simultaneously. 

The  tooth  outlines  that  are  generated  simultaneously  are  the 
parts  of  the  teeth  that  are  in  contact,  which  is  the  face  of  one 
tooth  with  the  flank  of  its  mate  on  the  other  gear. 

If  the  addendum  circles  are  as  shown  and  the  upper  gear  is  the 
driver  turning  in  a  counter  clockwise  direction,  contact  will  begin 
at  a  where  the  addendum  circle  of  the  driven  cuts  the  rolling 
circle  of  the  driver.  It  will  follow  along  the  upper  rolling  circle 
until  the  pitch  point  is  reached.  At  this  point  contact  will 
change  from  the  drivers  flank  and  followers  face,  to  the  drivers 
face  and  followers  flank  where  it  will  follow  along  the  rolling 
circle  of  the  driven  to  the  point  b  where  the  addendum  circle 
of  the  driver  cuts  the  rolling  circle  of  the  driven,  where  contact 
between  that  pair  of  teeth  will  cease. 

From  the  figure  it  can  be  seen  that  the  length  of  the  face  of 
the  driver's  teeth  governs  the  angle  of  recess,  and  the  length  of 
the  face  of  the  driven  gear's  teeth  governs  the  angle  of  approach. 

Thus,  if  no  angle  of  approach  is  desired,  the  teeth  of  the  driven 
gear  will  have  no  faces,  and  the  driver's  teeth  will  require  no 
flanks.  This  case  is  sometimes  used  in  the  case  of  the  rack  and 
pinion  on  planer  beds. 

It  will  also  be  noted  that  if  the  diameters  of  the  rolling  circles 
are  made  equal  to  the  pitch  radii  of  their  respective  gears,  that 
the  flanks  of  the  teeth  will  be  radial. 

105.  Cycloidal  Rack  and  Pinion. — In  the  rack  and  pinion 
combination  shown  in  Fig.  105  let  AP  be  the  radius  of  the  pitch 
circle  of  the  pinion,  and  C  and  D  the  centers  of  the  rolling  circles. , 

The  upper  rolling  circle  when  rolled  on  the  inside  of  the  pitch 
circle  of  the  pinion  generates  the  hypocycloid  Pm,  the  flank  of 
the  pinion  teeth,  and  when  rolled  on  the  pitch  line  of  the  rack 
generates  the  cycloid  Ps,  the  face  of  the  rack  teeth.  The  lower 


104 


MECHANISM 


rolling  circle  when  rolled  on  the  outside  of  the  pinion's  pitch 
circle  generates  the  epicycloid  Pr}  the  face  of  the  pinion  teeth, 
and  generates  the  flank  of  the  rack  teeth  when  rolled  on  the  lower 
side  of  the  pitch  line  of  the  rack,  which  is  the  cycloid  Pn. 

If  the  pinion  drives  clockwise  as  shown,  contact  will  begin  at 
a  on  the  rolling  circle  of  the  pinion,  and  cease  at  6  on  the  rolling 
circle  of  the  rack. 


FIG.  105. 

106.  Cycloidal  Rack  Teeth  with  Straight  Flanks. — In  order 
to  have  straight  flanks  on  the  rack  teeth,  the  rolling  circle  of 
the  rack  will  be  of  infinite  diameter  and  will  coincide  with  the 
pitch  line.     The  faces  of  the  pinion  teeth,  which  are  developed 
by  this  circle,  will  therefore  be  involutes.     The  disadvantage 
of  this  form  of  tooth  is  that  all  of  the  wear  on  the  rack  teeth  is 
between  the  pitch  and  addendum  lines,  there  being  no  contact 
on  the  rack  teeth  below  the  pitch  line. 

107.  Cycloidal  Annular  Gear  and  Pinion. — In  Fig.  106  let  AP 
be  the  pitch  radius  of  the  pinion  and  BP  the  pitch  radius  of 
the  annular  gear  in  an  annular  gear  and  pinion  combination. 
Let  PC  be  the  radius  of  the  inside  rolling  circle,  and  PD  the 


GEARING  FOR  PARALLEL  SHAFTS 


105 


radius  of  the  outside  rolling  circle.  The  outside  rolling  circle 
will  generate  the  epicycloids  Pm  and  Pn  which  are  the  flank  of 
the  annular  gear  teeth  and  the  face  of  the  pinion  teeth  respec- 
tively. The  face  of  the  annular  gear  tooth  is  the  hypocycloid 
Pr,  while  the  flank  of  the  pinion  tooth  is  the  hypocycloid  Ps. 
These  curves  are  generated  by  the  inside  rolling  circle. 


With  the  pinion  driving  as  shown,  contact  will  begin  at  a  and 
cease  at  6.  The  method  of  laying  out  the  gears  is  in  every 
respect  similar  to  that  of  spur  gears. 

108.  Limiting  Size  of  Annular  Gear  Pinion. — It  was  seen  in 
Art.  101  that  the  same  epicycloid  could  be  generated  by  two 
different  sized  rolling  circles.  The  difference  in  their  diameters 
being  equal  to  the  diameter  of  the  directing  circle.  On  account 
of  this  fact,  the  difference  between  the  pitch  circles  of  a  cycloidal 
annular  gear  and  pinion  combination  must  be  equal  to  the  sum 
of  the  diameters  of  the  rolling  circles  or  else  there  will  be  what  is 
called  secondary  contact  between  the  teeth.1 

If  the  pitch  circles  differ  by  the  sum  of  the  diameters  of  the 
rolling  circles,  and  the  gears  are  standard,  interchangeable  gears, 
they  will  differ  by  twelve  teeth. 

1  For  a  full  discussion  of  this  property  of  the  cycloidal  annular  gear  and 
pinion  see  Professor  MacCord's  "Kinematics,"  also  a  "Treatise  on  Gear 
Wheels,"  by  George  B.  Grant,  M.  E. 


106 


MECHANISM 


Provided  one  of  the  gears  have  no  faces  on  its  teeth,  they  need 
differ  by  but  six  teeth.1 

109.  Interchangeable  Cycloidal  Gears. — In  the  different  pairs 
Of  cycloidal  gears  thus  far  considered,  the  rolling  circles  have 
not  been  taken  of  the  same  diameter,  and  it  is  possible  to 
lay  out  the  teeth  of  two  gears  having  the  same  pitch  that 
will  mesh  properly  together  even  though  their  rolling  circles 
are  not  the  same  size.  If,  however,  the  gears  are  to  be  in- 
terchangeable, that  is,  any  gear  mesh  with  any  other  of  the 
set,  they  must,  besides  having  the  same  pitch,  have  their  tooth 
curves  developed  by  the  same  size  rolling  circles.  In  Fig.  107  if 


the  gears  A,  B  and  C  are  all  of  the  same  pitch,  and  their  tooth 
curves  developed  by  rolling  circles  of  the  diameters  shown,  B 
will  mesh  properly  with  A  and  (7,  but  A  and  C  will  not  mesh 
with  each  other. 

110.  Standard  Diameter  of  Rolling  Circle  for  Cycloidal  Gears. — 
The  standard  diameter  of  rolling  circle  is  one  that  will  give 
radial  flanks  on  a  gear  of  twelve  teeth.  In  order  to  have  radial 
flanks  on  the  teeth,  the  diameter  of  the  rolling  circle  must  be 
one-half  that  of  the  pitch  circle.  (Art.  102). 

This  does  not  mean  that  all  the  gears  of  the  set  will  have  radial 
flanks,  but  only  the  12-tooth  gear. 

For  example,  the  standard  diameter  rolling  circle  for  a  64-tooth 
4-pitch  gear  would  be  the  same  as  that  for  a  12-tooth  of  the  same 
pitch,  or  J^-  =  3-in.  pitch  diameter,  and  f  in.  =  lj-in.  diameter  of 
rolling  circle. 

Geo.  B.  Grant  says:2  "The  standard  adopted  by  the  manu- 

1  The  reason  for  this  is  discussed  in  Art.  110. 
a  Treatise  on  Gear  Wheels,  page  41. 


GEARING  FOR  PARALLEL  SHAFTS  107 

facturers  of  cycloidal  gear  cutters  is  that  having  radial  flanks  on 
a  gear  of  fifteen  teeth,  but  is  not  and  should  not  be  in  use  for 
other  purposes.  If  any  change  is  made,  it  should  be  in  the 
other  direction,  to  make  the  set  take  in  gears  of  ten  teeth. 

"It  must  be  borne  in  mind  that  the  standard  adopted  does  not 
limit  the  set  to  the  stated  minimum  number  of  teeth,  but  that  it 
simply  requires  that  the  smaller  gears  have  weak  under  curved 
teeth." 

111.  Comparison  of  Involute  and  Cycloidal  Systems. — In  the 
two  systems,  the  advantages  are  practically  all  in  favor  of  the 
involute  tooth.  Some  of  them  are  as  follows: 

1.  The  distance  between  the  centers  can  be  changed  without 
affecting  the  velocity  ratio. 

2.  The  path  of  contact  is  a  straight  line,  thus  keeping  a  con- 
stant pressure  on  the  axes. 

3.  Tooth  curves  of  single  curvature. 

4.  Involute  rack  teeth  have  straight  sides,  thus  cutter  is  easily 
made. 

5.  Wear  on  involute  teeth  more  uniform  than  in  cycloidal. 

6.  Less  cutters  required  to  cut  a  complete  set  of  gears. 

The  cycloidal  system  is  the  older  of  the  two,  and  for  those 
using  it,  the  expense  of  changing  over  to  the  involute  system 
would  be  large,  but  the  main  objection  would  be  that  for  repairs 
on  machines  already  equipped  with  cycloidal  gears,  cutters  of 
that  system  must  be  kept  on  hand. 

PROBLEMS 

51.  Construct  an  epicycloid  for  a  3  in.  generating  circle  and  an  8  in. 
directing  circle. 

52.  Construct  hypocycloids  using  a  12  in.  directing  circle  and  generating 
circles  of  4  in.  and  7  in.  respectively. 

53.  In  a  cycloidal  spur  gear  combination  of  unequal  size,  assume  the 
diameters  of  the  pitch,  addendum  and  rolling  circles,  and  show  the  path  of 
contact  for  the  small  gear  driving  clockwise. 

54.  In  a  cycloidal  rack  and  pinion  combination  with  the  pinion  driving 
counter-clockwise,    assume  the  diameters  of  pitch,  addendum  and  rolling 
circles  so  that  there  will  be  no  angle  of  recess,  and  show  path  of  contact. 

55.  What  are  the  standard  diameters  of  rolling  circles  that  would  be 
used  to  generate  the  tooth  outlines  for  each  of  the  following  gears? 

1.  28  teeth  2  pitch  4.  86  teeth  4  pitch. 

2.  49  teeth  2  pitch.  5.  4  pitch  20  in.  pitch  diameter. 

3.  100  teeth  12J  in.  pitch  diameter.     6.  50  teeth  1  in.  pitch. 


108  MECHANISM 

56.  Draw  the  correct  tooth  curves  for  two  spur  gears  that  are  in  mesh, 
cycloidal  system.     Cut  templets  to  lay  out  the  tooth  curves  and  check 
them  for  contact.     Show  the  teeth  going  into  and  out  of  action. 

Data:  Number  of  teeth  in  gears  36  and  21. 
Diametral  pitch  1J. 
Diameter  of  rolling  circles  4  in. 
Thickness  of  rim  below  tooth  bottoms  2  in. 
Small  gear  to  drive  and  revolve  clockwise. 

Note. — Make  center  line  of  gears  horizontal  and  in  center  of  the  sheet  with 
the  smaller  gear  to  the  right.  Layout  the  pitch,  addendum,  dedendum  and 
rolling  circles.  Develop  the  tooth  curves  and  cut  templets  to  fit  them. 
Space  out  the  teeth  on  the  pitch  circles.  Ink  drawing.  Time,  5  hours. 

57.  Draw  the  correct  tooth  curves  for  a  cycloidal  rack  and  pinion  that 
are  in  mesh.     Check  the  templets  for  contact  and  show  the  teeth  going 
into  and  out  of  contact. 

Data:  Teeth  in  pinion  21. 
Diametral  pitch  1£. 
Diameter  of  rolling  circles  4  in. 

Note. — Make  rack  horizontal  and  center  of  pinion  in  center  of  sheet  right 
and  left  and  1£  in.  from  top  border  line.  Pinion  to  drive  counter-clock- 
wise. Use  templet  of  21-tooth  gear  of  problem  56.  No  statement  of 
problem  but  this  title,  CYCLOIDAL  RACK  &  PINION.  Time,  4  hours. 

58.  Draw  the  correct  tooth  curves  for  a  cycloidal  annular  gear  and 
pinion  that  are  in  mesh. 

Data:  Same  as  for  problem  56.  Use  same  templets.  No  statement  of 
problem  or  data,  but  this  title,  CYCLOIDAL  ANNULAR  GEAR  &  PINION. 

Time,  3  hours. 

_ 

PIN  GEARS 

112.  Pin  Gearing. — If  the  diameter  of  the  rolling  circle  having 
its  center  at  D,  Fig.  104,  is  increased  until  it  coincides  with  its 
pitch  circle  it  will  generate  epicycloids  for  the  faces  of  the  teeth 
of  A  and  a  point  for  the  teeth  of  B. 

Since  a  point  or  pin  tooth  could  do  no  work,  it  is  usual  to 
substitute  a  pin  of  sensible  diameter,  then  the  tooth  curve  on 
the  other  gear  will  be  parallel  to  the  true  epicycloid,  and  a  dis- 
tance away  from  it  equal  to  the  radius  of  the  pin.  In  using  pin 
gears,  a  small  number  of  pins  can  be  used  on  a  gear  to  correspond 
to  a  small  number  of  teeth,  but  they  have  not  the  disadvantage 
of  having  weak  teeth.  Fig.  108  shows  a  pin  wheel  of  six  teeth 
meshing  with  a  toothed  gear.  The  dotted  outline  shows  the 
teeth  of  gear  if  the  pins  had  no  diameter.  The  circular  pitch 
must  of  course  be  the  same  in  each  gear,  or  the  arc  PI'  must 
equal  the  arc  PI.  The  maximum  radius  of  addendum  circle  is 


GEARING  FOR  PARALLEL  SHAFTS 


109 


found  by  drawing  a  straight  line  PI  and  finding  where  it  cuts 
the  circumference  of  the  pin  at  a.  The  radius  of  the  addendum 
circle  will  be  from  the  center  of  the  gear  to  a. 

Sometimes  rolls  are  placed  on  the  pins  in  order  to  make  the 
action  between  the  teeth  more  nearly  a  rolling  one.  In  this  case 
the  pins  are  secured  between  two  circular  plates,  and  the  gear 
is  then  known  as  a  lantern  wheel.  This  type  is  common  in  watch 
and  clock  mechanisms,  except  that  no  rolls  are  used. 


\ 


FIG.  108. 


The  pins  should  always  be  on  the  driven  wheel  since  then  the 
action  between  the  teeth  is  almost  all  on  the  recess  side.  It  will 
be  wholly  so  if  the  pins  have  no  diameter. 

Pin  gears  are  not  confined  to  spur  gears,  but  can  be  used  in 
any  combination,  and  the  pins  can  be  on  either  gear,  as  a  pin 
annular  gear  and  a  toothed  pinion. 


STEPPED  AND  HELICAL  GEARS 

113.  Stepped  and  Helical  Gears. — Smoother  action  can  be 
obtained  in  a  pair  of  gears  by  having  more  teeth  in  contact  at 
one  time.  This  can  be  done  by  lengthening  the  teeth  or  by 
making  the  pitch  smaller.  By  lengthening  the  teeth,  the  angle 
of  action  is  made  greater  and  since  more  teeth  are  brought  into 
contact  at  one  time,  the  pressure  on  each  tooth  is  decreased,  but 


110 


MECHANISM 


gear  teeth  are  assumed  to  act  as  cantilever  beams  and  the  teeth 
are  lengthened  at  a  greater  rate  than  the  number  of  pairs  of 
teeth  in  contact  is  increased,  so  that  the  resulting  gear  is  weaker. 

By  decreasing  the  pitch,  the  thickness  of  the  tooth  is  decreased, 
and  this  is  at  a  greater  rate  than  the  pairs  of  teeth  are  increased, 
so  again  the  gear  is  made  weaker. 

More  pairs  of  teeth  can  be  brought  into  contact  at  one  time 
by  placing  several  gears  side  by  side  on  the  same  shaft,  and 
advancing  the  teeth  of  each  ahead  of  the  one  next  to  it,  by  the 
circular  pitch  divided  by  the  number  of  gears,  and  then  fastening 
them  securely  together.  Such  a  gear  is  called  a  stepped  gear. 


FIG.  109. 


FIG.  110. 


If  the  number  of  steps  is  increased  infinitely,  and  the  teeth  on 
each  placed  -  th  of  the  circular  pitch  ahead  of  the  one  next  to  it, 

where  n  equals  the  number  of  steps,  a  line  drawn  through  the 
center  of  the  top  of  each  tooth  will  be  a  helix,  and  such  gears 
are  called  helical,  spiral1  or  twisted  gears.  A  pair  of  such  gears 
are  shown  conventionally  in  Fig.  109. 

If  the  power  to  be  transmitted  is  great,  there  will  be  con- 

1  The  term  "spiral"  as  applied  to  gears,  suggests  a  gear  the  teeth  of  which 
lie  in  a  plane,  but  the  word  is  used  to  denote  gears,  the  teeth  of  which  have 
helical  form,  and  while  it  has  common  usage  the  terms,  twisted  or  helical 
would  perhaps  convey  a  better  idea  of  the  gear.  This  is  suggested  by  Mr. 
R.  E.  Flanders  in  his  book  on  "  Gear  Cutting  Machinery." 


GEARING  FOR  PARALLEL  SHAFTS 


111 


siderable  end  thrust,  that  is,  one  gear  will  tend  to  slide  by  the 
other  along  its  shaft.  This  tendency  is  overcome  by  placing  on 
the  shaft  gears  of  the  same  size  and  of  equal  but  opposite  helix 
angles.  These  gears  are  represented  conventionally  in  Fig.  110, 
and  are  known  as  "herring-bone"  gears. 

There  has  recently  been  introduced  into  this  country  a  system 
of  herring-bone  gears  known  as  the  Wuest  gear,  named  after  its 
inventor.  The  gear  is  cut  from  a  solid  piece  and  the  teeth  of  one 
half  are  advanced  one-half  the  circular  pitch  ahead  of  the  other.1 


APPROXIMATE  METHODS  OF  LAYING  OUT  TOOTH  CURVES 

114.  There  are  a  number  of  methods  of  laying  out  tooth  curves 
by  using  templets,  rectangular  coordinates  and  circular  arcs. 
It  is  not  necessary  to  use  these  methods  as  much  since  gear  cut- 
ting machinery  has  been  brought  to  such  a  high  state  of  develop- 
ment as  it  was  when  the  pattern  maker  or  millwright  had  to 
lay  out  the  teeth  of  gears  that  were  usually  cast. 


FIG.  111. 

The  Robinson  Templet  Odontograph  is  one  of  the  best  known 
templet  methods  for  laying  out  tooth  curves.  The  odontograph, 
together  with  the  method  of  forming  the  side  of  a  tooth,  is 
illustrated  in  Fig.  111.  It  is  necessary  to  have  a  table  to  set  the 
instrument  properly. 

1  See  article  on  "The  Herring-bone  Gear"  by  P.  C.  Day.  Journal  of  the 
American  Society  of  Mechanical  Engineers,  for  Jan.,  1912. 


112 


MECHANISM 


The  curves  of  the  sides  of  the  templet  are  logarithmic  spirals, 
one  being  the  evolute  of  the  other. 

The  holes  in  the  templet  are  for  the  purpose  of  fastening  it  to  a 
bar,  so  that  it  can  be  swung  about  the  center  of  the  gear  for  the 
purpose  of  drawing  in  all  of  the  tooth  outlines.  It  will  be  noticed 
that  two  settings  of  the  instrument  are  necessary  to  draw  in  one 
side  of  a  tooth.  The  Willis  Odontograph  is  illustrated  in  Fig. 
112  and  is  for  laying  out  involute  teeth  by  the  method  of  circular 


\FiG.  112. 

x'  • 

fS  x 

arcs.  The  two  legs  of/the?  templet' make  an  angle  of  90°  minus 
the  angle  of  obliquity  with  each  other.  The  setting  for  the 
radiu&'6f  the  tooth  outlinq  depends  upon  the  pitch  diameter  of 
the  gear,,  and  after  the  setting  is  obtained,  the  teeth  can  be 
drawn  in.  J 


A  =  Face  Radius. 
B=  Flank  Radius. 

FlG.   113. 


Grant's  Involute  Odontograph  is  also  a  method  of  laying  out 
involute  teeth  by  means  of  circular  arcs.  The  lengths  of  radii 
to  be  used  for  different  pitches  and  numbers  of  teeth  can  be 
found  from  Table  C. 

The  method  of  laying  out  the  radii  is  shown  in  Fig.  113. 


GEARING  FOR  PARALLEL  SHAFTS 


113 


TABLE  C.— GRANT'S  INVOLUTE  ODONTOGRAPH 
STANDARD  INTERCHANGEABLE  TEETH 
Centers  on  Base  Line 


Teeth 

Divide  by 
the  diametrical  pitch 

Multiply  by  the  circular  pitch 

Face  radius 

Flank  radius 

Face  radius 

i 

Flank  radius 

10 

2.28 

.69 

.73 

.22 

11 

2.40 

.83 

.76    : 

.27 

12 

2.51 

.96 

.80 

'    -31 

13 

2.62 

.09 

.83    | 

.34 

14 

2.72 

.22 

.87 

]     .39 

15 

2.82 

.34 

.90 

.43 

16 

2.92 

.46 

.93 

.47 

17  ' 

3.02 

.58 

.96 

.50 

18 

3.12 

1  .  69 

.99 

.54 

19 

3.22 

1.79 

1.03 

.57 

20 

3.32 

1.89 

1.06 

.60 

21 

3.41 

1.98 

t.Q9 

\       .63 

22 

3.49 

2.06 

1.11 

.66 

23 

3.57 

2.15 

1.13 

.69 

24 

3.64 

2.24 

1.16 

.71 

25 

3.71 

2.33 

1.18 

.74 

26 

3.78 

2.42 

1.20 

.77 

27 

3.85 

2.50 

1.23 

.80 

28 

3.92 

2.59 

1.25 

.82 

29 

3.99 

2.67 

1.27 

.85 

30 

4.06 

2.76 

1.29 

.88 

31 

4.13 

2.85 

1.31 

.91 

32                     4.20 

2.93 

1.34 

.93 

33 

4.27 

3.01 

1.36 

.96 

34 

4.33 

3.09 

1.38 

.99 

35 

4.39 

3.16 

1.39 

1.01 

36 

4.45 

3.23 

1.41 

1.03 

37-40 

4.20 

1.34 

41-45 

4.63 

1.48 

46-51 

5.06 

1.61 

52-60 

5.74 

1.83 

61-70 

6.52 

2.07 

71-90 

7.72 

2.46 

91-120 

.9.78 

3.11 

121-180 

13.33 

4.26 

181-360 

21.62 

6.88 

114 


MECHANISM 

GEAR  CUTTING 


115.  Cutting  Spur  and  Annular  Gears. — The  general  method 
for  cutting  these  gears  is  to  use  a  tool,  the  outline  of  the  cutting 
face  of  which  is  the  same  as  the  space  between  two  teeth.  Fig. 
114  shows  a  tool  that  can  be  used  in  a  planer  or  shaper  for 


b   c 


FIG.  114. 


FIG.  115. 


cutting  spur  and  annular  gears.  The  profile  abed  is  the  shape 
of  the  space  between  the  teeth.  The  tool  can  be  sharpened 
across  the  front  face  without  changing  the  profile. 

Fig.  115  shows  a  circular  cutter  for  use  in  a  milling  machine. 
This  also  has  the  profile  abed  the  shape  of  the  space  between 
the  teeth,  and  the  face  can  be  sharpened  parallel  to  the  axis 
without  changing  the  profile.  This  cutter  cannot  be  used  as 
readily  for  annular  gears  as  that  of  Fig.  114. 

116.  Interchangeable  Gear  Cutters. — In  order  to  use  either  of 
the  two  cutters  just  described  to  cut  a  gear  theoretically  correct, 
it  is  necessary  to  have  a  different  cutter  for  each  different  number 
of  teeth  and  pitch,  since  the  space  between  the  teeth  changes  with 
each  number  of  teeth.  Therefore  to  cut  a  complete  set  of  gears  of 
any  one  pitch,  from  twelve  teeth  to  a  rack  theoretically  correct, 
it  will  take  as  many  different  cutters  as  there  are  gears. 

It  has  been  found,  however,  that  one  cutter  can  be  used  to 
cut  several  different  numbers  of  teeth  of  any  one  pitch,  accurate 


GEARING  FOR  PARALLEL  SHAFTS 


115 


enough  for  practical  purposes,  as  in  the  larger  gears  the  space 
does  not  change  very  much  by  the  addition  of  a  few  teeth. 

The  following  table  shows  the  cutters  used  by  the  Brown  and 
Sharpe  Mfg.  Co.  for  gear  teeth. 


Involute  cutters 
8  cutters  in  each  set 


Cycloidal  cutters 
24  cutters  in  each  set 


Cutter 

Teeth 

Cutter         Teeth 

Cutter 

Teeth 

No.  1  cuts 

135  to  rack 

A  cuts 

12 

M  cuts 

27  to    29 

No.  2  cuts 

55  to  134 

B  cuts 

13 

N  cuts 

30  to    33 

No.  3  cuts 

35  to    54 

Ccuts 

14 

O  cuts 

34  to    37 

No.  4  cuts 

26  to    34 

D  cuts 

15 

Pcuts 

38  to    42 

No.  5  cuts 

21  to    25 

E  cuts 

16 

Q  cuts 

43  to    49 

No.  6  cuts 

17  to    20 

Fcuts 

17 

R  cuts 

50  to    59 

No.  7  cuts 

14  to    16 

G  cuts 

18 

Scuts 

60  to    74 

No.  8  cuts 

12  to    13 

Hcuts 

19 

T  cuts 

75  to    99 

I  cuts 

20 

U  cuts 

100  to  149 

J  cuts 

21  to  22 

Vcuts 

150  to  249 

Kcuts 

23  to  24 

Wcuts 

250  or   more 

L  cuts 

25  to  26 

X  cuts 

rack. 

The  cutters  are  given  in  terms  of  the  number  of  teeth,  for  by 
this  method  they  are  applicable  to  all  pitches. 

It  will  be  noticed  that  there  are  eight  cutters  necessary  for 
a  set  of  involute  gears,  while  in  the  cycloidal  system  there  are 
twenty-four  cutters.  The  reason  for  this  is  that  in  the  involute 
system  the  tooth  outlines  do  not  change  as  rapidly  as  in  the 
cycloidal  system. 

117.  Conjugate  Methods  of  Cutting  Teeth. — If  instead  of  using 
cutters  that  have  the  shape  of  the  space  between  the  teeth, 
those  having  an  outline  the  same  as  the  tooth  of  a  gear  are  used, 
it  is  possible  to  cut  a  complete  set  of  gears  of  one  pitch  with  a 
single  cutter. 

This  is  known  as  the  molding-generating  principle  and  is 
largely  used  for  cutting  spur  gear  teeth. 

Imagine  a  gear  blank  of  wax  or  other  plastic  material  rolled 
with  a  gear  or  rack  on  which  the  teeth  have  been  cut,  their 
pitch  circles  being  tangent  and  rolling  together  without  slipping. 


116 


MECHANISM 


Teeth  will  be  formed  in  the  soft  material  of  the  blank,  which  will 
mesh  properly  with  the  molding  gear. 

The  gear  blank  of  course  cannot  be  made  of  wax,  so  in  order 
to  get  the  same  result,  the  molding  gear  has  cutting  edges  on  its 
teeth  and  is  given  a  reciprocating  motion  across  the  face  of  the 
blank.  The  rolling  of  the  pitch  circles  together  taking  place 


after  the  cutting  gear  has  passed  across  the  face  of  the  blank 
and  come  back  ready  for  the  second  stroke. 

In  Fig.  116  let  A  be  a  single  tooth  of  an  involute  rack1  and  B 
a  blank  on  which  teeth  are  to  be  cut.  Let  the  motion  be  such 
that  the  pitch  line  of  the  rack  tooth  and  the  pitch  circle  of  the 
gear  roll  together  without  slipping,  and  also  give  the  rack  tooth, 
which  has  a  cutting  edge  like  the  planer  tool  of  Fig.  114,  a 


FIG.  117. 

reciprocating  motion  across  the  face  of  the  blank.  The  different 
positions  represent  the  stages  in  the  cutting  of  the  space  between 
two  teeth. 

All  of  the  gears  cut  with  this  tooth  will  mesh  properly  with  it 
and  will  also  mesh  correctly  with  each  other. 

1  Any  other  involute  or  cycloidal  gear  tooth  could  be  used  equally  well. 


GEARING  FOR  PARALLEL  SHAFTS 


117 


Fig.  117  shows  a  complete  gear  for  a  cutter  instead  of  a  single 
tooth.  The  gear  has  cutting  edges  the  same  as  the  single  tooth, 
and  is  first  fed  in  to  the  proper  depth,  that  is  so  that  the  pitch 
circles  are  tangent,  and  then  both  rotate  slightly  before  beginning 
each  cutting  stroke.  Several  teeth  are  in  process  of  being  cut 
at  one  time,  although  but  one  tooth  is  completed  at  a  time. 

118.  Fellows  Gear  Shaper. — This  machine,  illustrated  in  Fig, 
118,  uses  a  cutter  shaped  like  a  pinion,  and  works  on  the  molding- 
generating  principle. 


FIG.  118. 

The  cutter  and  blank  are  fastened  to  vertical  axes  which  are 
connected  so  that  when  one  rotates  the  other  is  rotated  the 
proper  amount,  the  two  moving  so  that  their  pitch  circles  roll 
together  without  slipping.  In  addition  to  this,  the  spindle 
carrying  the  cutter  can  reciprocate  in  the  direction  along  its 


118  MECHANISM 

axis,  so  that  when  ascending  the  teeth  of  the  cutter  act  as  cutting 
tools  on  the  blank  in  the  same  way  as  the  cutting  tool  of  a 
shaper. 

The  cutter  spindle,  which  is  carried  on  a  cross-rail  so  that  the 
axis  of  the  cutter  can  be  moved  toward  that  of  the  blank. 

The  action  of  the  machine  is  as  follows:  The  cutter  and  blank 
are  keyed  to  their  respective  axes,  with  neither  the  cutter  nor 
the  blank  rotating. 

During  the  up-stroke,  the  teeth  of  the  cutter  cut  metal  from 
the  blank,  and  on  the  down-stroke  the  cutter-spindle  is  fed 
toward  the  blank  axis  a  small  amount. 

This  feeding  in  motion  is  continued  at  the  end  of  each  non- 
cutting  stroke  until  the  pitch  circles  are  tangent  and  then  ceases. 
In  place  of  this,  at  the  end  of  each  non-cutting  stroke,  both  the 
cutter  and  blank  axes  rotate  slightly  through  corresponding 
angles  so  that  their  pitch  circles  roll  together  without  slipping. 

During  the  cutting  stroke  neither  the  cutter  nor  blank  rotate. 
After  the  blank  has  made  one  complete  revolution  the  teeth  are 
all  cut.  Each  tooth  of  the  cutter  acts  as  a  cutting  tool  in  turn, 
although  several  teeth  of  the  blank  are  acted  upon  at  one  time. 

One  cutter  will  cut  all  gears  of  the  same  pitch,  and  all  gears 
that  are  cut  by  the  cutter  will  mesh  properly  with  it,  and  with 
each  other. 

Either  involute  or  cycloidal  cutters  can  be  used.  An  objec- 
tion to  the  cycloidal  form  of  cutter  is  that  they  must  be  pro- 
duced by  means  of  templets,  while  the  involute  form  can  actu- 
ally be  generated. 

119.  Gear  Hobbing. — The  gear  hobbing  machine  is  generally 
designed  to  cut  teeth  by  the  molding-generating  principle  in 
which  a  rack  form  of  tooth  is  used  for  the  cutter,  instead  of  the 
circular  cutter,  but  instead  of  a  single  rack  tooth  as  in  Fig.  116, 
a  'hob*  is  used.  A  hob  is  a  cutter  shaped  like  a  worm,  with  its 
sides  gashed  similarly  to  a  reamer  or  tap  to  form  cutting  faces. 
Fig.  119  illustrates  the  principle,  but  instead  of  a  hob,  an  ordi- 
nary worm  is  shown.  In  a  section  of  the  worm  along  its  axis  the 
teeth  have  straight  sides,  the  same  as  involute  rack  teeth,  the 
rack  being  shown  in  dotted  outline. 

The  teeth  of  the  worm  or  hob,  being  of  helical  form,  like  a 
screw  thread,  its  axis,  in  order  to  cut  teeth  on  the  blank  parallel 
with  the  axis  of  the  blank,  cannot  be  perpendicular  to  the  blank, 
but  must  be  at  an  angle  of  90°  minus  the  helix  angle  of  the  worm. 


GEARING  FOR  PARALLEL  SHAFTS 


119 


FIG.  119. 


120 


MECHANISM 


Fig.  120  shows  a  gear  bobbing  attachment  on  a  universal  mill- 
ing machine,  in  which  the  hob  is  placed  on  the  arbor  of  the 
machine  with  the  hob  and  blank  driven  at  the  desired  velocity 
ratio. 

The  hob  is  slowly  fed  across  the  face  of  the  blank,  and  when 
it  has  gone  all  the  way  across  the  work  is  completed. 


FIG.  120. 


CHAPTER  VIII 
BEVEL  GEARS,  WORM  AND  WORM  WHEEL 

120.  In  all  of  the  gears  thus  far  considered,  the  elements  of 
the  teeth  were  parallel  to  each  other,  and  with  the  exception  of 
the  helical  gears,  the  elements  of  the  teeth  were  parallel  to  the 
axis  of  the  gear,  the  tooth  outlines  being  generated  by  a  right 
line  which  was  the  element  of  a  flexible  band  as  in  the  involute 
system,  or  the  element  of  a  rolling  cylinder  in  cycloidal  system. 

In  bevel  gears,  the  axes  intersect,  the  pitch  cylinders  becom- 
ing pitch  cones,  and  the  elements  of  the  teeth  converging  at  the 
point  of  intersection  of  the  shafts. 


FIG.  121. 

An  idea  of  this  can  be  had  if  it  is  imagined  that  alternate 
grooves  and  projections  are  placed  on  the  friction  cones  of  Fig.  83 
as  was  done  in  Figs.  86  and  87.  The  cones  will  then  look  as  in 
Fig.  121,  which  shows  a  pair  of  bevel  gears. 

The  pitch  cones  are  segments  of  a  sphere,  and  in  order  to 
lay  out  the  gears  theoretically  correct,  the  tooth  outlines  need 
to  be  laid  out  on  the  surface  of  the  sphere,  making  use  of  spherical 
trigonometry.  With  the  exception  of  this  difference,  the  bevel 
gear  differs  little  in  theory  from  the  ordinary  spur  gear. 

121 


122  MECHANISM 

Bevel  gears  are  always  laid  out  in  pairs;  and  are  not  inter- 
changeable as  are  gears  for  parallel  axes. 

121.  Velocity  Ratio. — The  velocity  ratio  for  bevel  gears  is 
inversely  as  the  radii  of  the  bases  of  their  pitch  cones,  and  the 
method  of  laying  out  a  pair  of  cones  for  any  desired  velocity 
ratio  was  discussed  in  Art.  84. 

122.  Miter  Gears. — These  are  bevel  gears  that  are  of  the  same 
size  and  in  which  the  shaft  angle  is  90°.     The  velocity  ratio  in  a 
pair  of  miter  gears  is  therefore  as  1 :1. 

123.  Crown  Gears. — A  crown  gear  is  one  in  which  the  sides 
of  the  pitch  cone  make  an  angle  of  180°  with  each  other.     The 
base  of  the  pitch  cone  is  therefore  a  great  circle  of  the  sphere 
on  which  the  teeth  of  the  crown  gear  is  developed.     The  crown 
gear  is  sometimes  called  the  rack  of  bevel  gears. 

The  involute  crown  gear  has  not,  however,  teeth  with  straight 
sides  as  in  the  involute  rack  of  spur  gears.  The  crown  gear  with 
straight  sided  teeth  is  known  as  the  octoid  tooth,  so  called  from 
the  peculiar  figure  eight  line  of  action.  This  tooth  is  the  inven- 
tion of  Mr.  Hugo  Bilgram  of  Philadelphia,  and  will  be  referred 
to  again  under  bevel  gear  cutting. 

LAYING  OUT  THE  TEETH  OF  BEVEL  GEARS 

124.  Tredgold's  Method. — It  was  stated  above  that  to  be  laid 
out  theoretically  accurate,  bevel  gears  must  be  laid  out  on  a 
spherical  surface,  and  as  this  is  not  developable  it  is  necessary 
to  use  an  approximate  method,  and  the  one  most  commonly 
employed  is  known  as  Tredgold's  method. 

Let  OA  and  OB,  Fig.  122,  be  the  axes  of  two  cones  COP  and 
POD  respectively,  their  bases  being  a  spherical  surface.  Normal 
to  the  elements  of  these  cones,  draw  the  cones  PAC  and  PBD, 
the  bases  of  these  second  or  "back"  cones  coinciding  with  those 
of  the  cones  first  drawn.  If  these  back  cones  are  cut  along  one 
of  their  elements  and  rolled  out  into  the  plane  of  the  paper,  the 
radius  of  the  upper  one,  the  apex  of  which  is  at  A,  will  be  AP, 
and  the  length  of  the  arc  PE  =  PCXn.  The  lower  cone  will 
have  a  radius  BP,  and  the  length  of  the  arc  PF  =  PDXn.  On 
the  arcs  PE  and  PF  can  be  laid  out  the  tooth  curves,  the  same 
as  for  spur  gears  having  pitch  radii  AP  and  BP.  The  pitch 
must  be  such  that  the  gears  will  have  a  whole  number  of  teeth. 
After  laying  out  the  tooth  curves  on  the  developed  back  cones, 


BEVEL  GEARS 


123 


the  cones  can  be  rolled  back  into  their  original  positions  and  the 
teeth  drawn  on  the  pitch  cones. 

It  will  be  noticed  that  that  part  of  the  back  cones  on  which  the 
teeth  are  laid  out  differs  very  little  from  the  spherical  surface. 


Fig.  123  shows  one  of  these  cones  laid  out  with  half  of  the 
teeth  drawn  in.  The  teeth  in  this  figure  appear  in  their  true 
thickness  in  the  plan  view,  but  are  foreshortened. 

The  method  of  construction  is  clearly  shown  in  the  figure  so 
that  there  is  no  need  of  further  explanation. 

125.  Shop  Drawing  of  Bevel  Gears. — In  making  a  drawing 
for  use  in  the  shop,  it  is  not  necessary  to  go  into  the  detail  of 
drawing  in  the  tooth  curves  as  in  the  previous  article,  but  a  much 
more  simple  drawing  together  with  the  necessary  angles  and 
dimensions  is  all  that  is  required. 

Problem. — Make  a  complete  working  drawing  of  a  pair  of  bevel 
gears  of  26  and  20  teeth;  diametral  pitch  4;  involute  system; 


124 


MECHANISM 


in.    diameter  of  shafts  1  in. 


shaft  angle  90°,  width  of  face 
diameter  of  hubs  2J  in. 

Make  axis  of  large  gear  horizontal. 

26  teeth,  4  pitch  =  6  J  in.  pitch  diameter. 

20  teeth,  4  pitch  =  5  in.  pitch  diameter. 

Draw  the  axes  of  the  gears  OA  and  OB,  Fig.  124,  making  an 
angle  of  90°  with  each  other.  From  0  on  OA  lay  off  2J  in.,  the 
pitch  radius  of  the  small  gear  and  draw  CP  perpendicular  to  OA. 


FIG.  123. 

On  OB  from  0  lay  off  3 J  in.  the  pitch  radius  of  the  large  gear  and 
draw  PD  perpendicular  to  OB.  Complete  the  pitch  cones  POC 
and  POD  and  draw  the  back  cones,  or  at  least  that  part  of  them 
where  the  teeth  are  to  be  laid  out. 

The  angles  E  and  Ef  are  called  the  pitch,  or  center  angles. 

From  the  tables  of  tooth  parts  find  the  addendum  and  deden- 
dum  of  the  teeth  and  lay  them  out  from  C,  P  and  D  as  shown. 
From  the  points  thus  found,  draw  toward  the  apex  of  the  pitch 
cones  and  lay  off  the  face  of  the  gears. 

The  angles  that  the  addendum  and  dedendum  lines  make 


BEVEL  GEARS 


125 


with  the  axes  of  the  gears,  give  the  face  and  cutting  angles  F 
and  C  respectively. 

The  angle  of  the  edge,  is  T,  and  is  the  same  as  the  center  or 
pitch  angle. 

The  angles  U  and  V  in  the  small  figure  are  called,  respectively, 
the  angle  increment  and  angle  decrement.  These  are  the  differ- 


FIG.  124. 


ences  between  the  pitch  angle  and  the  face  and  cutting  angles 
respectively. 

When  the  axes  of  the  gears  are  at  right  angles,  the  various 
angles  and  the  outside  diameter  can  be  found  as  follows: 

In  the  small  figure, 

Let  We  =  pitch  radius  of  small  gear 

Let  OW  =  pitch  radius  of  large  gear 


126  MECHANISM 

m       _.      We     pitch  radius  of  small  gear 

Inen  Ian  Jb  =  „„,  —    ..  , r. ^ = 

OW      pitch  radius  of  large  gear 

number  of  teeth  in  small  gear 
number  of  teeth  in  large  gear 
We 


OC  = 


sin  E 


Then  Tan  U=-%-  and  Tan  7= 


Oc  ~  Oc 

cb  and  cd  are  the  face  and  flank  of  the  tooth,  the  dimensions 
being  found  in  the  table  of  tooth  parts. 

The  angle  acb  =  angle  E. 

Then  ac  =  cb  cos  acb 

.*.  Pitch  diameter  of  large  gear  -f  2ac  =  outside  diameter  of 
large  gear. 

In  a  similar  manner  the  distance  R  can  be  found. 

It  is  of  course  necessary  to  find  these  angles  and  dimensions 
for  each  of  the  gears. 

When  the  shaft  angle  is  not  90°  the  method  is  not  quite  as 
simple,  but  after  finding  the  pitch  angles,  the  others  are  found 
in  a  manner  similar  to  that  described.  By  some  authorities,  R 
is  called  the  "backing,"  while  others  use  H  as  the  backing. 

The  other  dimensions  necessary  for  a  complete  shop  drawing 
are  indicated  in  the  figure. 

CUTTING  BEVEL-GEAR  TEETH 

126.  Approximate  Methods. — Bevel-gear  teeth  being  on  the  sur- 
face of  a  cone,  their  section  is  different  at  all  points  along  the 
elements  of  the  face.  It  is  not  possible  therefore  to  cut  the  teeth 
theoretically  correct  with  circular  cutters.  For  narrow  faces  it 
can  be  done  approximately  by  using  a  cutter  the  shape  of  the 
tooth  at  the  large  end,  but  about  0.005  in.  less  than  the  thick- 
ness of  the  space  at  the  small  end,  measured  on  the  pitch  line. 
The  settings  of  the  milling  table  and  blank  are  so  made  that 
one  side  of  all  the  teeth  is  finished  first,  then  with  another  setting 
of  the  table,  the  other  side  is  milled. 

Sometimes  for  gears  that  are  to  be  cut  with  circular  cutters, 
the  blank  is  turned  up  so  that  the  addendum  line  of  the  face  of 
the  gear  is  parallel  with  the  pitch  line,  or  the  face  of  the  cone 
instead  of  being  drawn  to  the  apex  of  the  pitch  cone  is  drawn 


BEVEL  GEARS 


127 


parallel  with  the  pitch  cone.  In  cutting  the  teeth  on  such  a 
blank  their  depth  is  made  constant. 

Mr.  C.  H/ Logue  says,1  "The  milling  of  bevel  gears  is  now 
practically  a  thing  of  the  past,  except  for  an  occasional  job 
required  for  shop  use,  the  generating  of  bevel  gears  having 
reached  such  a  point  that  the  milling  machine  cannot  compete 
either  in  quality  or  time." 

127.  Accurately  Cut  Bevel-gear  Teeth.— Fig.  125  illustrates 
diagramatically  a  method  of  cutting  the  teeth  of  bevel  gears  in 
which  a  former  or  templet  is  used. 


FIG.  125. 

The  templet  is  made  the  shape  of  the  teeth  that  are  to  be  cut, 
and  being  farther  from  the  apex  of  the  pitch  cone,  is  made  to  a 
larger  scale  than  the  tooth  outline.  The  cutting  tool  is  carried 
on  a  frame,  the  back  end  of  which  is  supported  by  a  roll  resting 
on  the  templet. 

The  cutting  point  of  the  tool  moves  with  a  reciprocating 
motion  along  the  line  OA,  which  is  tangent  to  the  templet  and 
roll,  and  passes  through  the  apex  of  the  pitch  cone. 

Another  method  of  guiding  the  rear  end  of  the  cutting  tool 
frame  is  to  move  it  in  a  circular  arc  by  means  of  a  link,  instead 
of  by  using  a  templet.  This  is  called  the  odontograph  method. 

128.  Bilgram  Bevel-gear  Planer. — This  machine  bears  the 
same  relation  to  the  bevel  gear  that  the  Fellows  gear  planer  does 
to  the  spur  gear,  in  that  the  teeth  are  generated.  In  Art.  117 

1  American  Machinist  Gear  Book,  page  153. 


128 


MECHANISM 


was  discussed  a  method  of  making  interchangeable  gears  by 
means  of  a  rack  tooth,  and  all  gears  that  would  mesh  correctly 
with  the  rack  would  also  mesh  correctly  with  each  other.  In 
the  Bilgram  bevel-gear  plaiier,  the  cutting  tool  corresponds  to 
the  straight-sided  tooth  of  an  octoid  crown  gear. 

The  space  between  the  teeth  of  bevel  gears  being  of  varying 
width,  however,  it  is  not  possible  to  use  a  tool  the  shape  of  Fig. 
116,  but  only  the  right  or  left  half  of  it. 


FIG.  126. 

The  planer  illustrated  in  Fig.  126  is  automatic,  and  consists  of 
two  principal  parts,  the  shaper  which  holds  and  operates  the 
tool,  and  the  evolver  which  holds  and  moves  the  blank. 

The  blank  must  move  as  a  rolling  cone,  which  is  done  first  by 
securing  its  arbor  in  an  inclined  position  to  a  semi-circular 
horizontal  plate  between  two  uprights.  This  semi-circular  plate 
can  be  oscillated  on  a  vertical  axis  passing  through  the  apex  of 


BEVEL  GEARS 


129 


the  blank,  and  is  driven  by  a  worm  and  worm-wheel  combination. 
The  second  part  of  the  motion  is  obtained  by  means  of  two  steel 
bands  stretched  in  opposite  directions,  one  end  of  each  being 
fastened  to  the  frame  and  the  other  ends  wrapped  around  a 
cone,  which  corresponds  to  the  pitch  cone  of  the  blank. 

The  tool  which  is  operated  by  a  Whitworth  quick-return 
motion,  cuts  on  the  forward  stroke,  and  during  each  return 
stroke  the  tool  is  raised  clear  of  the  blank  by  means  of  a  cam, 
and  the  blank  rotated  one  tooth  by  means  of  gearing  and  index 
mechanism.  All  spaces  are  treated  in  the  same  manner,  after 
which  the  tool  automatically  adjusts  itself  for  the  second  cut  on 
a  tooth,  and  so  on  until  the  blank  is  completed.  A  similar  tool 
is  used  to  finish  the  other  sides  of  the  teeth. 

The  inclination  of  the  arbor  that  holds  the  blank  is  made 
adjustable  in  order  to  adapt  it  to  the  angle  of  the  desired  gear. 
This  adjustment  must  be  exactly  concentric  with  the  apex  of 
the  blank.  Ordinarily  a  different  sized  rolling  cone  would  have 
to  be  used  to  cut  each  different  sized  gear,  but  means  have  been 
devised  whereby  a  limited  number  of  cones  are  sufficient.1 

PROBLEMS 

59.  Make  a  finished  pencil  or  ink  shop  drawing  of  a  pair  of  bevel  gears  of 
36  teeth  and  27  teeth  respectively,  3  pitch  and  15°  angle  of  obliquity.  Face 
of  gears  2£  in. 


Note. — Put  on  all  of  dimensions  and  angles  shown  in  Fig.  124.  No  state- 
ment of  problem  to  go  on  sheet,  but  this  data.  BEVEL  GEARS 

Teeth    36  and  27 
Pitch    3  diametral 
Involute    System 
Time,  4  hours. 

60.  Make  a  sketch  showing  how  you  would  find  the  pitch  cones  for  a  pair 
of  bevel  gears  in  which  the  driver  made  five  revolutions  to  seven  of  the 
driven,  with  a  115°  shaft  angle. 

1  For  a  more  complete  description  of  this  machine  see  American  Machinist 
of  May  9,  1885,  and  Jan.  23,  1902. 


130 


MECHANISM 


WORM  AND  WORM  WHEEL 

129.  When  two  shafts  are  at  right  angles  but  do  not  intersect, 
motion  can  be  transmitted  between  them  by  means  of  a  worm 
and  worm-wheel  combination,  which  is  illustrated  in  Fig.  127. 

Velocity  Ratio. — Worm  gearing  differs  from  spur  gearing  in 
that  the  velocity  ratio  does  not  depend  upon  the  diameters  of 
the  gears,  but  is  the  relation  between  the  number  of  threads  on 
the  worm  (single  or  multiple)  and  the  number  of  teeth  in  the 
worm  wheel.  The  worm  is  always  the  driver,  and  for  a  single 


FIG.  127. 

thread  worm,  each  revolution  of  the  worm  would  rotate  the  worm 
wheel'  one  tooth,  while  if  it  were  a  double  thread  worm,  the 
worm  wheel  would  be  rotated  two  teeth  and  so  on.  For  ex- 
ample, with  a  single  thread  worm  and  a  48-tooth  wheel,  it  would 
require  48  revolutions  of  the  worm  to  rotate  the  wheel  once. 
If  the  worm  were  of  double  thread,  it  would  require  24  revolutions 
of  the  .worm  for  one  of  the  wheel,  while  if  a  triple  thread  worm 
were  used  16  revolutions  of  the  worm  would  be  necessary  for  one 
of  the  wheel. 

Thus  it  will  be  seen  that  the  pitch  diameters  do  not  enter  into 
the  question. 

If  the  pitch  and  number  of  teeth  of  the  wheel  are  given,  and 
also  the  distance  between  the  shaft  centers,  the  pitch  diameter  of 
the  worm  should  be  such  that  the  two  pitch  circles  are  tangent. 


BEVEL  GEARS 


131 


130.  Pitch. — The  circular  pitch  system  is  used  in  making 
worm  and  wheel  calculations,  for  the  reason  that  the  worms  are 
usually  cut  in  the  lathe  the  same  as  screw  threads,  and  lathes 
are  not  equipped  with  the  proper  change  gears  for  cutting  di- 
ametral pitches. 

The  objection  to  the  circular  pitch  system  to  spur  gears, 
that  the  distance  between  the  shaft  centers  will  be  an  incon- 
venient fraction  unless  the  circular  pitch  is  inconvenient,  does 
not  apply  in  this  case  however. 


>f  Woi 


FIG.  128. 

The  involute  system  is  used  in  worm  gearing  for  the  reason 
that  the  worm  will  have  teeth  of  straight  sides  which  are  easily 
cut  in  the  lathe. 

Fig.  128  shows  two  views  of  a  worm  and  wheel,  the  left-hand 
view  being  a  section  along  the  axis  of  the  wheel  and  the  right- 


132 


MECHANISM 


hand  view  a  section  along  the  axis  of  the  worm.  From  these 
views  it  can  be  seen  that  the  section  of  the  worm  in  the  right-hand 
view  is  the  same  as  an  involute  rack  and  the  section  of  the  wheel 
(not  wholly  cross-hatched),  in  the  same  view  is  the  same  as  that 
of  a  spur  gear  having  the  same  pitch  diameter  and  number  of 
teeth  as  the  wheel.  The  throat  diameter  of  the  wheel  is  the  same 
as  the  outside  diameter  of  a  spur  gear  having  the  same  pitch 
and  number  of  teeth. 

131.  Contact  between  the  worm  and  wheel  teeth  is  not  very 
definitely  determined  but  it  is  generally  believed  to  be  nearly 
line  contact  between  any  pair  of  teeth  for  small  gears,  although 
it  may  have  some  width,  and  becomes  surface  contact  for  the 
larger  gears.  To  get  as  much  contact  as  possible,  the  worm  wheel 
is  constructed  so  that  its  sides  partially  envelop  the  worm.  The 
included  angle  between  the  sides  is  generally  made  60  degrees  or 
90  degrees. 


600 

CAA 

I 

Worm  Wheel 
5  1  1  1  g  \ 

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D12345678910111S 
Worm-Threads  Long 

FIG.  129. 

132.  Length  of  Worm. — The  worm  is  not  limited  in  length  but 
should  be  long  enough  to  provide  all  of  the  contact  that  can  be 
obtained  between  any  pair  of  teeth.  It  is  often  made  longer 
than  necessary  and  moved  along  the  shaft  when  it  becomes  worn, 
to  bring  a  new  part  of  the  worm  in  contact  with  the  wheel  teeth. 

The  curve1  shown  in  Fig.  129  gives  the  length  of  worm  required 
to  get  the  maximum  contact  when  14^-degree  angle  of  obliquity 

1  From  data  of  Brown  and  Sharpe  Manufacturing  Company. 


BEVEL  GEARS  133 

is  used.  Contact  will  not  be  along  the  whole  length  of  the  worm 
on  its  axial  section,  but  on  the  sides  as  the  wheel  partially 
envelops  the  worm. 

133.  Cutting  the  Worm-wheel  Teeth. — A  worm  is  first  made  in 
the  lathe,  the  same  as  the  worm  that  is  to  mesh  with  the  wheel, 
except  that  its  diameter  is  twice  the  clearance  greater,  then  its 
sides  are  fluted  similar  to  a  reamer  or  tap  in  order  to  form 
cutting  edges.     This  hob  and  the  wheel  blank  are  then  mounted 
in  a  milling  machine,  the  hob  being  fed  down  into  the  blank 
until  their  pitch  lines  are  tangent,  then  both  are  turned  at  the 
proper  velocity  ratio.     The  only  difference  between  hobbing  the 
worm  wheel  and  the  spur  gear  shown  in  Fig.  120  is  that  in  the 
worm  wheel  the  axis  of  the  hob  is  directly  over  the  center  line 
of  the  blank,  and  does  not  move  across  it. 

For  worm  wheels  of  less  than  30  teeth  the  flanks  will  be  under- 
cut unless  the  blank  is  made  over-size. 

If  the  axis  of  the  worm  deviates  from  a  right  angle  with  that 
of  the  wheel  by  an  amount  equal  to  the  thread  angle  of  the  worm, 
the  worm  can  be  used  with  an  ordinary  spur  gear.  This  is 
sometimes  done  in  rough  work. 

ELLIPTICAL  GEARS 

134.  Elliptical  gears  are  the  most  common  of  the  non-circular 
gears. 


FIG.  130. 

In  order  that  two  ellipses  roll  together  properly,  they  must  be 
equal  and  similar,  and  each  ellipse  must  rotate  about  one  of  its 
foci,  the  distance  between  the  centers  of  rotation  being  equal  to 
the  major  axis. 

Let  A  and  B,  Fig.  130,  be  two  ellipses  with  foci  ab  and  cd 


134 


MECHANISM 


respectively.  The  links  joining  opposite  foci  form  a  crossed 
link  kinematic  chain  similar  to  Fig.  15  in  which  the  crossed  and 
opposite  links  are  equal.  The  point  of  intersection  of  the  crossed 
links  is  at  the  tangent  point  P,  of  the  ellipses.  If  one  of  the 
ellipses  is  held  stationary,  the  other  can  be  rolled  with  pure 
rolling  contact  around  it. 

Velocity  Ratio. — Since  the  two  ellipses  are  the  same  size,  the 
velocity  ratio  will  be  as  1  : 1,  but  at  any  part  of  the  revolution  the 
ratio  is  inversely  as  the  radii  from  the  fixed  foci  to  the  tangent 
point  P}  of  the  ellipses. 

Any  system  of  teeth  that  is  practicable  for  spur  gears  can 
be  used  for  elliptical  gears  and  the  teeth  if  cut  with  the  same 
cutter  will  all  have  different  profiles.  This  may  result  in  weak 
under-cut  teeth  at  and  near  the  ends  of  the  ellipses,  but  can  be 
remedied  by  using  one  cutter  for  the  ends  where  the  ellipse  has  a 
small  radius,  and  another  for  the  sides  where  the  radius  is  larger. 
The  blanks  are  usually  cut  by  clamping  them  together  and  cutting 
both  at  one  operation.  This  insures  the  mating  teeth  being  the 
same. 

Elliptical  gears  can  be  used  for  "quick-return  motions"  such 
as  slotters  and  shapers,  where  all  of  the  work  is  done  on  half 
of  the  revolution,  the  other  half  being  used  to  get  the  tool  out 
of  the  way  and  the  work  ready  for  the  next  stroke. 

They  are  not  in  very  general  use  as  they  are  hard  to  cut 
without  the  use  of  special  attachments. 

PROBLEMS 

61.  Make  a  full-size  working  drawing  of  a  worm  and  worm-wheel  com- 
bination. Involute  system. 


Key 


Data:     Velocity  ratio  32  : 1. 

Single  thread  right-hand  worm. 
Circular  pitch  |  in. 


BEVEL  GEARS  135 

Note. — Make  teeth  in  contact  at  pitch  point.  Draw  tooth  outlines  of 
wheel  teeth  by  means  of  Grant's  Involute  Odontograph  table.  Develop 
helix  for  outline  of  worm  teeth.  No  statement  of  problem,  but  this  data, 

WORM  &WORM  WHEEL. 
Teeth  in  wheel  32 
Single  thread  R.H.  worm 
Circular  Pitch  £  in. 
Involute  System 

Time,  4  hours. 


CHAPTER  IX 
GEAR  TRAINS 

135.  When  more  than  two  gears  are  in  mesh,  such  a   com- 
bination is  a  train  of  mechanism  (Art.  16),  and  is  called  a  gear 
train  or  train  of  gears. 

It  is  often  desirable  to  find  the  revolutions  of  the  last  gear  of 
a  train  without  finding  the  revolutions  of  each  intermediate  gear. 

The  number  of  teeth  on  any  two  gears  that  mesh  together  is 
proportional  to  their  respective  pitch  diameters,  and  since  the 
revolutions  are  inversely  proportional  to  the  pitch  diameters 
(Art.  15),  therefore,  their  revolutions  are  also  inversely  pro- 
portional to  the  number  of  teeth  on  the  gears. 

136.  Value  of  a  Train. — By  this  expression  is  meant  the  ratio 
of  the  revolutions  of  the  first  and  last  gear  of  the  train  in  a 
given  time. 


FIG.  131. 

In  Fig.  131  let  A,  B,  C,  D,  E,  and  F  be  the  pitch  circles  of 
gears  on  the  shafts  1,  2,  3,  4  and  5,  and  having  the  number  of 
teeth  as  shown. 

The  gear  A  drives  B,  B  drives  C,  and  since  C  and  D  are  on  the 
same  shaft  they  will  rotate  as  one  gear.  D  drives  E  and  E 
drives  F. 

Revolutions  of  B  _  80.  Revolutions  of  C_20 
Revolutions  of  A  20*  Revolutions  of  B  70 
Revolutions  of  E _25.  Revolutions  of  F _40 
Revolutions  of  D  40'  Revolutions  of  E  50 
Revolutions  of  F _80  20  25  40_4 
Ur'  Revolutions  of  A~20X70X40X50~7 

136 


GEAR  TRAINS  137 

That  is  for  each  revolution  of  A,  F  will  make  y  of  a  revolution. 
Instead  of  the  number  of  teeth,  the  pitch  diameters  can  be  used. 
A  rule  for  the  above  may  be  expressed  as  follows:  To  find  the 
number  of  revolutions  of  the  last  gear  of  a  train,  divide  the  con- 
tinued products  of  the  teeth  or  pitch  diameters  of  all  the  drivers 
times  the  revolutions  of  the  first  gear,  by  the  continued  products 
of  the  teeth  or  pitch  diameters  of  all  the  driven  gears. 

It  is  not  necessary  that  the  same  unit  be  used  throughout  the 
various  ratios,  but  the  same  unit  must  be  used  in  the  same  ratio. 
That  is,  we  may  have  a  60-tooth  gear  driving  a  45-tooth  gear  and 
a  12-in.  gear  driving  a  6-in.  gear,  and  so  on,  but  we  cannot  use  a 
60-tooth  gear  driving  a  12-in.  gear. 

A  gear  train  may  be  written  in  the  following  manner  to  avoid 
making  a  sketch  of  the  gears. 

First    axis,    64    teeth. 

Second  axis,  32 -10" 

Third  axis,  15-50  teeth. 

Fourth  axis,  20 

Fifth  axis,  30-12"  (annular). 

Sixth  axis,  3  —  24  tooth. 

All  of  the  gears  on  the  same  horizontal  line  are  on  the  same  shaft, 
and  each  gear  drives  the  one  directly  below  it. 

If  the  first  gear  of  this  train  makes  60  revolutions  per  minute, 
^        i        t^          -       MTU    64X10X50X20X12X60      C001 
the  value  of  the  tram  will  be       32xl5x2oX30X3       =533i 

revolutions  per  minute  for  the  sixth  axis. 

137.  Idle  Gear  and  Direction  of  Rotation. — An  idle  gear  is  a 
gear  placed  between  two  axes  for  the  purpose  of  changing  the 
directional  relation  of  the  two.  Thus  if  two  gears  are  placed  in 
direct  contact  their  direction  of  rotation  will  be  opposite,  but  if 
one  intermediate  gear  is  placed  between  them  their  direction  of 
rotation  will  be  the  same.  The  idler  does  not  change  the  velocity 
ratio,  since  it  is  both  a  driver  and  a  driven.  In  Fig.  131,  the  gears 
B  and  E  are  idlers  and  the  velocity  ratio  would  not  be  affected 
if  they  were  left  out.  The  gears  C  and  D  however  are  not  idlers 
although  they  are  on  the  same  shaft,  since  the  number  of  teeth 
on  them  is  not  the  same. 

In  spur  gears  when  there  is  an  odd  number  of  intermediate 
axes  between  the  first  and  last,  the  direction  of  rotation  of  the 


138 


MECHANISM 


first  and  last  axes  will  be  the  same,  while  if  there  is  an  even  num- 
ber of  intermediate  axes,  the  direction  of  rotation  will  be  opposite. 

If  there  is  one  annular  gear  in  the  train,  it  changes  the  direc- 
tion of  rotation  of  the  last  gear  from  what  it  would  be  otherwise. 
Thus  in  the  last  problem  worked  out  there  are  four  intermediate 
axes  and  one  annular  gear,  so  that  the  direction  of  rotation  of  the 
first  and  last  axes  will  be  the  same. 

138.  Gear  Train  for  Thread  Cutting.— Fig.  132  illustrates  the 
gearing  found  on  an  ordinary  engine  lathe  that  is  equipped  for 
cutting  screw  threads.  On  the  spindle  E,  are  the  gears  F,  C  and 


FIG.  132. 

D  and  the  stepped  cone.  The  gear  D  is  keyed  to  the  spindle  and 
the  stepped  cone  is  free  to  revolve  on  the  spindle  except  when  it 
is  locked  to  D  by  means  of  a  pin. 

The  gear  C  is  fastened  to  the  stepped  cone  and  revolves  with  it. 
A  and  B  are  the  back  gears  and  are  keyed  to  a  shaft  that  is  paral- 
lel to  the  spindle.  This  shaft  has  eccentric  bearings  so  that  the 
back  gears  may  be  thrown  out  of  mesh  with  C  and  D. 

When  the  back  gears  are  "thrown  out,"  D  and  the  stepped 
cone  are  locked  together,  and  as  many  speeds  can  be  obtained  as 
there  are  steps  on  the  cone,  while  with  the  back  gears  "thrown 


GEAR  TRAINS  139 

in,"  D  and  the  stepped  cone  are  not  locked  and  the  number  of 
speeds  of  the  spindle  is  doubled. 

F  is  the  spindle  gear  and  is  for  driving  the  thread-cutting  train. 
G  and  H  are  tumbling  gears,  or  tumblers  and  are  for  the  purpose 
of  changing  the  direction  of  rotation  of  the  screw  M. 

G  meshes  with  7  which  is  the  inside  stud  gear,  and  the  arrange- 
ment is  such  that  F  and  H  may  be  thrown  out  of  mesh  and  G  be 
made  to  mesh  directly  with  F.  F  and  7  are  generally  made  the 
same  size,  but  when  they  are  not  the  same,  7  is  usually  made 
twice  the  size  of  F,  so  that  7  makes  one-half  as  many  revolutions 
as  the  spindle. 

On  the  same  shaft  with  7  is  the  outside  stud  gear  /.  Meshing 
with  J  is  an  intermediate  K  which  meshes  with  the  screw  gear  L 
on  the  end  of  the  lead  screw  M .  The  only  gears  that  it  is  neces- 
sary to  change  in  thread  cutting  are  the  stud  gear/  and  the  screw 
gear  L.  The  intermediate  K  is  held  on  a  slotted  bracket,  not 
shown  in  the  figure,  which  allows  K  to  be  adjusted  to  accommo- 
date different  sized  gears  J  and  L. 

The  cutting  tool  is  held  in  a  tool  holder  which  is  moved  back 
and  forth  along  the  work  by  means  of  a  split  nut  engaging  the 
lead  screw. 

If  there  are  eight  threads  per  inch  on  the  lead  screw,  then  eight 
revolutions  of  the  lead  screw  would  advance  the  cutting  tool 
one  in.,  and  if  the  work  made  four  revolutions  to  eight  of  ths 
lead  screw,  four  threads  per  inch  would  be  cut  on  the  work. 

To  find  the  number  of  teeth  in  the  stud  and  screw  gears  when 
F  and  7  are  of  the  same  size.  Since  the  gears  G,  H  and  K  are 
idlers  they  need  not  enter  into  the  calculations,  and  since  the 
spindle  and  stud  shafts  make  the  same  number  of  revolutions  it 
is  only  necessary  to  find  the  revolutions  of  the  stud  shaft. 

Let  JV  =  number  of  threads  per  inch  to  be  cut  on  work. 
Let   n  =  number  of  threads  per  inch  on  lead  screw. 
Let  T  =  number  of  teeth  on  screw  gear. 
Let    t=  number  of  teeth  on  stud  gear. 

Revolutions  of  stud  shaft  _  Number  of  teeth  on  screw  gear 
Revolutions  of  lead  screw"  Number  of  teeth  on  stud  gear 

or  -  =  y  and  T  =  -X  t. 

n      t  n 

The  following  table  of  change  gears  was  taken  from  the  plate 
on  a  16  in.  swing  lathe,  having  six  threads  per  inch  on  the  lead 


140  MECHANISM 

screw  and  twice  as  many  teeth  on  the  inside  stud  gear  as  on  the 
spindle  gear. 


Threads 

Teeth  on 

Teeth  on 

Threads 

Teeth  on 

Teeth  on 

to  be  cut 

stud  gear 

screw  gear 

to  be  cut 

stud  gear 

screw  gear 

3 

80 

20 

15 

48 

60 

4 

72 

24 

16 

48 

64 

5 

48 

20 

18 

48 

72 

6 

48 

24 

20 

24 

40 

7 

48 

28 

22 

24 

44 

8 

48 

32 

24 

24 

48 

9 

48 

36 

26 

24 

52 

10 

48 

40 

28 

24 

56 

11 

48 

44 

30 

24 

60 

Hi 

48 

46 

32 

24 

64 

12 

48 

48 

36 

24 

72 

13 

48 

52 

40 

24 

80 

14 

48 

56 

48 

20 

80 

To  obtain  a  greater  range  of  threads  without  adding  more 
change  gears,  the  intermediate  gear  K,  on  Fig.  132,  might  be 
compounded,  that  is,  replaced  by  two  gears  of  different  sizes,  and 
fastened  together,  one  of  them  meshing  with  the  stud  gear  and 
the  other  with  the  screw  gear. 

The  involute  system  is  used  for  change  gear  sets  for  in  this 
system  the  center  distance  can  be  varied  without  affecting  the 
velocity  ratio. 

PROBLEMS 

62.  How  many  r.p.m.  does  the  last  gear  of  the  following  train  make,  if  the 
first  gear  makes  25  r.p.m.?     Is  its  direction  of  rotation  the  same  or  opposite? 

12  in. 
6  in. 

20  in.     -  6  in. 
12  in. 

20  in.  -48T  (annular) 
247 -12  in. 

63.  The  first  gear   of  the  following  train  makes  100  r.p.m.  and  turns 
clockwise.     How  many  r.p.m.  does  the  last  gear  make? 

60T-4P. 

15  in. 

10  in.  -12  in. 

15  in.—  — 3077  (bevel) 

36Z7  (bevel) — single  Thd  worm 

12077  worm  wheel — 10  in. 

64.  Design  a  lathe  screw  train  for  right-  and  left-hand  thread  cutting,  to 
cut  8,  9,  10,  11,  11},  12,  13,  14,  16,  18,  20,  24,  and  27  threads  per  inch. 


GEAR  TRAINS  141 

Data:  Threads  per  inch  on  lead  screw  8. 

Teeth  in  spindle  and  inside  stud  gears  24. 

Teeth  in  tumblers  21. 

Teeth  in  intermediate  108. 

All  gears  12  pitch. 

Note.— Make  a  full  size  pencil  drawing,  representing  the  extreme  positions 
of  the  gears  by  their  pitch  circles. 


CHAPTER  X 
BELTING 

139.  Belts  are  flexible  connectors  and  are  used  for  connecting 
shafts  where  the  distance  is  too  great  for  gears,  or  where  an 
absolutely  constant  velocity  ratio  is  not  required. 

Belting  may  be  divided  into  two  general  classes,  flat  and  round; 
the  former  is  used  on  pulleys  with  faces  that  are  cylindrical  or 
nearly  so,  and  the  latter  which  is  usually  either  of  fiber  or  metal 
requiring  pulleys  with  grooves  or  flanges  to  keep  the  rope  from 
running  off  the  pulley  or  drum. 

140.  Velocity  Ratio  and    Directional    Rotation. — When    two 
pulleys  are  connected  by  an  inelastic  belt  of  no  appreciable  thick- 
ness, all  parts  of  the  belt  and  the  surfaces  of  the  pulleys  have  the 
same  linear  velocity,  if  there  be  no  slipping  and  therefore  their 
revolutions  are  inversely  proportional  to  their  radii. 

This  condition  is  not  attained  in  practice  on  account  of  the 
elasticity  and  thickness  of  the  belt.  In  order  to  be  accurate  the 
thickness  of  the  belt  should  be  added  to  the  diameters  of  the 
pulleys  in  making  calculations.  In  this  discussion,  where  the 
diameter  of  the  pulley  is  referred  to,  it  means  diameter  of  pulley 
plus  thickness  of  belt. 

When  the  velocity  ratio  between  two  shafts  is  large,  instead  of 
making  a  single  reduction,  it  may  be  made  in  one  or  more  steps 
by  using  a  counter  shaft  or  jack  shaft.  This  is  an  intermediate 
shaft  on  which  there  are  two  pulleys  of  different  diameters.  Thus 
if  the  velocity  ratio  between  two  shafts  were  1  : 40,  the  ratio 
might  be  divided  up  1  : 5  and  1  : 8. 

The  method  of  connection  shown  in  Fig.  133  is  an  open  belt, 
while  that  shown  in  Fig.  134  is  a  crossed  belt. 

With  an  open  belt  both  pulleys  turn  in  the  same  direction, 
or  the  directional  relation  is  the  -same,  while  in  the  case  of  the 
crossed  belt,  the  directional  relation  is  opposite. 

In  order  that  a  belt  may  maintain  its  position  on  a  pulley, 
its  center  line  on  the  approaching1  side  must  lie  in  a  plane  per  pen- 

1  By  the  approaching  side  is  meant  the  part  of  the  belt  advancing  toward 
the  pulley.  The  side  where  the  belt  leaves  the  pulley  is  called  the  receding 
side. 

142 


BELTING 


143 


dicular  to  the  axis  of  the  pulley.  If  a  force  is  applied  to  the  approach- 
ing side  of  the  belt  parallel  to  the  axis  of  the  pulley,  each  succeed- 
ing part  of  the  belt  will  take  up  a  position  a  little  farther  along  on 
the  face  of  the  pulley.  A  force  applied  to  the  receding  side  of 
the  belt  will  have  no  effect  on  that  pulley  unless  it  is  great 
enough  to  move  the  belt  bodily.  However  the  receding  side  with 
respect  to  one  pulley  is  the  approaching  side  with  respect  to  the 
other  pulley. 


FIG.  133. 


FIG.  134. 


141.  Crowning  Pulleys. — Because  of  the  fact  that  belts  do  not 
remain  exactly  straight  and  the  pulleys  in  exact  alignment,  some 
means  must  be  provided  for  keeping  the  belts  from  running  off, 
and  this  can  be  accomplished  by  means  of  flanges  on  the  pulleys, 
or  a  fork  placed  on  each  side  of  the  belt,  but  the  most  common 
method  is  to  crown  the  pulley.  By  this  is  meant  that  the  diame- 
ter of  the  pulley  is  greater  in  the  center  than  at  the  edges.  The 
crown  can  be  either  straight  or  spherical  and  its  amount  will 
depend  upon  the  width  and  speed  of  the  belt,  and  the  distance 
between  the  centers  of  the  pulleys.  Less  crown  is  required  the 


144 


MECHANISM 


wider  the  face  of  the  pulley,  the  greater  the  speed  and  the  greater 
the  distance  between  the  pulley  centers. 

142.  Tight  and  Loose  Pulleys. — When  machines  are  driven 
from  a  line  shaft,  and  it  is  not  desirable  to  have  them  run  all  of 
the  time,  means  must  be  provided  so  that  any  machine  may  be 
stopped  without  stopping  the  line  shaft.  Common  methods  of 
doing  this  are  by  means  of  clutches,  or  the  tight  and  loose  pulley 
combination.  The  tight  and  loose  pulleys  are  two  pulleys  placed 
side  by  side  on  the  counter  shaft,  the  tight  pulley  being  keyed  to 
the  shaft,  and  revolving  with  it,  while  the  loose  pulley  is  held  in 
place  by  a  collar,  and  is  free  to  revolve  on  the  shaft.  When  the 
machine  is  thrown  out  of  operation  the  belt  is  shifted  from  the 
tight  pulley  to  the  loose  pulley.  The  loose  pulley  should  be 
made  of  slightly  smaller  diameter  so  that  the  belt  will  not  be 
stretched  as  tightly  when  running  idle. 

Ceiling  '  


Counter  Shaft 


Hanger 


Head 


,  Floor 


Tail  Stock 


FIG.  135. 

Fig.  135  illustrates  a  method  of  connecting  a  lathe  with 
the  line  shaft  by  means  of  belts. 

The  pulley  on  the  line  shaft,  over  which  the  belt  runs  from  the 
tight  or  loose  pulley  is  made  with  a  cylindrical  face,  and  a  width 
at  least  equal  to  that  of  the  tight  and  loose  pulleys  combined. 

143.  In  Fig.  136  is  shown  a  method  of  connecting  two  shafts 
with  a  belt  when  they  are  too  close  together  to  connect  them  by 
the  methods  of  Figs,  133  or  134. 


BELTING 


145 


FIG.  136. 


FIG.  137. 


146 


MECHANISM 


144.  Belts  for  Intersecting  Shafts. — When  two  intersecting 
shafts  are  to  be  connected    by  belts,  guide  pulleys  must  be  so 
placed  that  the  belt  will  run  on  the  pulley  straight  (Art.  140). 
This  is  accomplished  in  Fig.  137  by  using  guide  pulleys  CC,  the 
axes  of  which  are  perpendicular  to  a  tangent  to  the  pulleys  A  and 
B  shown  in  the  upper  view,  and  their  faces  are  tangent  to  the 
center  line  of  the  pulleys  A  and  B  in  the  lower  view. 

145 .  Belts  for  Shafts  that  are  neither  Parallel  nor  Intersecting. — 
The  most  common  form  of  this  arrangement  is  the  "quarter 
turn"  belt  in  which  the  shaft  angle  is  90  degrees. 


FIG.  138 


FIG.  139. 


Fig.  138  shows  two  views  of  such  a  combination.  The  main 
point  to  bear  in  mind  when  adjusting  the  pulleys  for  a  belt  of 
this  kind  is  to  have  it  run  on  the  pulley  straight.  This  is  accom- 
plished if  a  plane  passed  through  the  middle  of  the  face  of  each 
pulley,  and  perpendicular  to  its  axis  is  tangent  to  the  face  of  the 
other  pulley.  It  will  be  noted  that  after  the  pulleys  are  adjusted, 
the  belt  will  run  in  one  direction  only.  In  order  to  have  it  run 
in  the  opposite  direction,  each  pulley  must  be  moved  along  its 
shaft  equal  to  the  diameter  of  the  other,  or  else  a  guide  pulley 
must  be  used  as  shown  in  Fig.  139. 

146.  Length  of  Belts. — If  the  distance  between  the  centers  of 
two  shafts  and  the  diameters  of  the  pulleys  are  known,  the  length 
of  belt  required  for  them  can  be  calculated  as  follows: 

1.  For  Open  Belts.     See  Fig.  140. 


BELTING 


147 


Let  L  =  distance  between  pulley  centers  in  inches. 
Let  R  =  radius  of  large  pulley  in  inches. 
Let  r    =  radius  of  small  pulley  in  inches. 

The  total  length  of  belt  is  the  length  not  in  contact  with  either 
pulley  plus  the  length  in  contact  with  the  large  pulley  plus  the 
length  in  contact  with  the  small  pulley. 


FIG.  140. 

Let  C  and  D  be  the  tangent  points  of  the  belt  and  pulleys. 
Then  2  CD  =  length  of  belt  not  in  contact  with  the  pulleys. 

2  CD  =  2\/L2-(R-r)2 
Length  of  belt  in  contact  with  large  pulley 


Length  of  belt  in  contact  with  small  pulley 

=  r(x-2d)=r  (x-2sm-i?j 
.'.Total  length  of  open  belt 


sn 


-r    sn 


- 


-     sn 


For  large  distances  between  shafts  and  with  pulleys  of  nearly 
the  same  diameter,  the  third  part  of  the  above  equation  may 
be  omitted. 


148 


MECHANISM 


2.  For  Crossed  Belts.     See  Fig.  141. 

Using  the  same  notation  as  before,  the  tangent  points  of  the 
belt  and  the  pulleys  are  C  and  D,  then  2CD  =  the  length  of  belt 
not  in  contact  with  the  pulleys. 

2CD  =  2VL2-(R+r)2 
Length  of  belt  in  contact  with  large  pulley 


Length  of  belt  in  contact  with  small  pulley 


.'.  Total  length  of  belt 


sn 


sn 


+r   x  +  2  sin 


FIG.  141. 


It  will  be  noticed  that  for  open  belts,  the  sum  and  difference  of 
the  radii  of  the  pulleys  are  used,  while  in  the  case  of  crossed  belts, 
only  the  sum  of  the  radii  are  considered. 

147.  Stepped  Cones  for  Same  Length  of  Belt.  —  On  most  machine 
tools  the  driving  pulley  is  usually  stepped,  the  belt  running  over 
a  similar  one  on  the  counter  shaft,  and  as  the  same  belt  must  be 
used  on  the  different  pairs  of  steps  they  must  be  designed  with 
this  in  mind. 

1.  For  Open  Belts.  —  No  simple  formula  has  been  derived  that 
can  be  used  for  calculating  the  size  of  steps  on  which  an  open 
belt  is  to  be  used,  so  this  problem  is  generally  solved  graphically. 


BELTING 


149 


An  approximate  method  often  used  is  that  of  C.  A.  Smith1  and 
is  as  follows: 

Let  A  and  B,  Fig.  142  be  the  centers  of  the  shafts,  and  L  the 
distance  between  them  in  inches. 

At  C  half  way  between  A  and  B}  erect  a  perpendicular  CD  of 
length  equal  to  0.314  L;  (this  coefficient  was  determined  experi- 
mentally2). 


FIG.  142. 

Draw  a  line  EF  tangent  to  the  pulleys  at  E  and  F.  With  D 
as  a  center  draw  the  arc  mn  of  such  a  radius  that  it  will  be  tangent 
to  the  line  EF. 

Let  BJ  be  the  radius  of  one  of  the  pulleys  of  the  second  pair 
of  steps;  to  find  its  mate  draw  the  line  JG  tangent  to  the  arc 
mn  and  tangent  to  an  arc  of  radius  BJ.  From  A  drop  a  per- 
pendicular AK  to  this  line  which  will  be  the  radius  of  the  mate. 

If  instead  of  the  diameter  of  one  of  the  second  pair  of  steps,  the 
velocity  ratio  of  the  second  pair  together  with  the  diameters  of 
the  first  pair  is  given,  the  solution  will  be  as  follows: 

Let  the  distance  from  where  the  line  JK  extended,  intersects 
the  line  of  centers  at  G,  to  A  equal  x.  Let  the  velocity  ratio  of 
A  =  .BJ  ^n 
B~ 


AK 
Then  by  similar  triangles 


x+L 
x 
L 


BJ 
AK 


x  = 


a-l 


So  that  by  taking  GA  =x  =  — — :  and  drawing  a  Ime  through  G 

tangent  to  FJ,  pulleys  for  the  required  velocity  ratio  will  be 
obtained. 

1  See  Transactions  American  Society  of  Mechanical  Engineers,  Vol.  X, 
page  269. 

2  When  the  angle  between  the  belt  and  center  line  of  pulley  exceeds  18 
degrees,  CD  is  taken  as  0.298  L. 


150  MECHANISM 

When  the  smaller  pulley  is  at  B,  the  value  of  a  becomes  less 
than  1,  and  that  of  x  negative,  which  indicates  that  it  must  be 
laid  off  from  A  toward  the  right  in  the  figure. 

2.  For  Crossed  Belts. — In  designing  stepped  pulleys  for  a 
crossed  belt,  it  is  only  necessary  to  keep  the  sum  of  the  radii  for 
the  different  pairs  of  steps  constant.  Thus  if  the  radii  of  one 
pair  of  steps  is  18  in.  each,  their  sum  is  36  in.  and  the  radii  of 
the  other  pairs  of  steps  that  would  take  the  same  length  of  crossed 
could  be  24  in.  and  12  in.,  30  in.  and  6  in.  and  other  radii  in 
which  their  sum  is  36  in. 

ROPE  DRIVING 

148.  In  this  form  of  belting,  the  pulleys  must  be  provided  with 
grooves  to  guide  the  rope.  For  fibrous  ropes  these  grooves  are 
generally  of  V  section,  the  rope  pulling  down  into  them  and 
increasing  the  tractive  power. 


FIG.  143. 


For  metal  rope  or  cable,  this  form  of  groove  would  be  injurious 
to  the  cable,  so  it  is  made  of  semi-circular  section,  the  cable 
resting  on  the  bottom  of  the  groove  which  may  have  a  leather, 
gutta-percha  or  wood  insert  to  prevent  metal  rubbing  on  metal. 


BELTING  151 

There  are  two  systems  of  rope  driving  in  general  use,  the 
English  or  individual  rope  system,  which  has  a  separate  rope 
for  each  groove  and  the  American  or  single  rope  system  in  which 
a  single  rope  is  carried  around  all  of  the  grooves  and  brought 
back  from  the  last  to  the  first  by  means  of  a  guide  pulley  set  at 
an  angle  with  the  other  pulleys. 

These  two  systems  are  shown  diagrammatically  in  Figs.  143 
and  144.  With  the  individual  ropes,  a  single  rope  may  break 
and  the  load  be  carried  by  the  others,  which  is  not  the  case  with 
the  single  rope  system.  On  the  other  hand,  it  is  difficult  to 
obtain  uniform  tension  on  all  of  the  ropes.  This  is  accomplished 
in  the  single  rope  system  by  means  of  a  tension  pulley  and  weight, 
so  that  the  wear  on  the  rope  is  more  uniform  throughout  its 
length  than  on  the  separate  ropes  of  the  individual  system. 


First 
Floor 


FIG.  144. 


CHAIN  DRIVING 

149.  The  chain  drive  is  used  generally  where  the  work  is 
heavy  and  a  positive  motion  is  required  at  a  comparatively  low 
speed.  The  wheels  over  which  the  chains  run  have  teeth  to 
receive  alternate  links  of  the  chain  and  are  called  sprocket  wheels. 
The  wheels  shown  in  Figs.  145  and  146  have  alternate  single  and 
double  teeth  to  receive  the  chain,  in  other  cases  the  teeth  of  the 
wheels  are  all  alike  as  in  bicycle  sprocket  wheels. 

In  Fig.  145  the  curve  of  the  tooth  may  be  found  by  taking  a 


152 


MECHAN.ISM 


center  at  A  and  a  radius  AB  minus  the  radius  of  the  end  of  the 
chain  link. 

To  reduce  friction  the  pins  are  often  provided  with  rolls  as  in 
pin  gearing.     This  form  of  chain  is  used  in  automobile  drives. 


FIG.  146. 

To  find  the  pitch  diameter  of  an  ordinary  sprocket  wheel, 
Let  P',  Fig.  147  be  the  pitch  or  length  of  one  link  of  the  chain.1 
Let  T=  number  of  teeth  in  wheel. 
Let  c  =  pitch  radius  of  wheel. 

1  This  can  be  readily  found  by  taking  a  piece  of  chain,  measuring  its 
length  and  dividing  by  the  number  of  links. 


BELTING 

Let  b  =  one-half  of  the  chordal  pitch. 


153 


360° 


x,         . 
then  sin 


=  -;  and  c  = 


After  chains  have  been  used  for  some  time,  the  pitch  may  change 
on  account  of  the  links  stretching  and  the  pins  becoming  worn, 
so  that  if  the  chain  originally  fitted  the  teeth,  it  will  not  later 
and  the  strain  may  be  all  carried  on  a  single  tooth. 


FIG.  147. 


FIG.  148. 


FIG.  149. 

150.  Renold  Silent  Chain. — This  is  the  invention  of  Hans 
Renold  and  is  a  great  improvement  in  chain  drives. 

A  short  section  of  the  chain  and  sprocket  is  shown  in  Fig.  148. 

The  chain  is  built  up  of  flat  pieces  of  which  lap  by  each  other 
and  are  fastened  together  by  steel  pins.  The  sides  of  the  teeth 
that  come  in  contact  with  the  wheel  as  well  as  those  of  the  wheel 
are  made  straight. 

It  is  very  satisfactory  for  motor  drives  where  a  positive  motion 
is  desired  and  the  distance  between  the  shafts  too  small  for 
belting.  It  also  always  fits  the  wheel  and  takes  up  the  backlash. 


154  MECHANISM 

151.  Morse  Rocker  Joint  Chain. — This  chain  shown  in  Fig.  149 
like  the  one  just  described,  is  built  up  of  flat  pieces  of  steel  but 
of  a  different  shape.  The  pins  instead  of  being  a  single  round  pin 
consist  of  two  hardened  steel  parts  so  formed  that  they  will  rock 
together  and  reduce  friction.  This  chain  can  be  used  in  place 
of  belting  for  speeds  up  to  2000  ft.  per  minute. 

PROBLEMS 

65.  A  countershaft  makes  50  r.p.m.  and  has  on  it  a  stepped  cone,  the 
diameters  of  the  steps  being  4  in.,  6  in.,  and  8  in.  respectively.     The  diame- 
ters of  the  corresponding  steps  on  the  machine  spindle  are  9  in.,  7  in.  and 
5  in.  respectively.     The  number  of  teeth  in  the  back  gears  are  40  and  25, 
and  the  gears  that  mesh  with  them  have  25  and  35  teeth.     How  many 
changes  of  speed  can  be  obtained,  and  what  will  be  the  r.p.m.  of  the  work 
for  each? 

66.  Calculate  the  lengths  of  crossed  and  open  belt  required  for  two  pulleys 
of  10  ft.  and  4  ft.  in  diameter  respectively,  and  25  ft.  between  centers. 

67.  An  emery  wheel  8  in.  diameter  is  to  have  a  peripheral  speed  of  2500 
ft.  per  minute.     It  is  driven  by  means  of  a  countershaft  upon  which  there 
are  two  pulleys,  one  18  in.  diameter  driven  by  means  of  a  belt  from  a  pulley 
36  in.  diameter  on  the  main  shaft,  and  the  second  which  drives  a  pulley 
2  in.  diameter  on  the  emery-wheel  shaft.     What  must  be  the  diameter  of 
the  second  pulley  on  the  countershaft  if  the  lineshaft  makes  80  r.p.m.? 

68.  Design  a  pair  of  stepped  pulleys  for  a  <  >    belt   in   which 

\    open    J 

the  distance  between  the  centers  is  24  in.,  diameter  of  smallest  step  4  in., 
diameter  of  shafts  1  in.  Revolution  per  minute  of  driver  200.  Revolutions 
per  minute  of  driven  to  be  50,  100,  200,  400,  and  800,  respectively  for  each 
of  the  pairs  of  steps. 

Note. — Lay  out  the  pulleys  according  to  the  formulas  for  crossed  belt  and 
by  Smith's  graphical  method  for  open  belts.  Check  the  diameters  of  each 
pair  of  pulleys  by  calculating  the  length  of  belt  required.  Make  the 
drawing  half  size.  Time,  4  hours. 


CHAPTER  XI 
INTERMITTENT  MOTIONS 

RATCHET  GEARING 

152.  Ratchet  Gearing. — This  is  perhaps  the  most  common 
form  of  intermittent  motion  and  consists  of  two  principal  parts, 
the  ratchet  wheel  and  pawl  or  detent. 

A  simple  form  is  shown  in  Fig.  150.  The  center  of  the  ratchet 
wheel  is  at  0,  and  the  driving  pawl  has  a  center  at  A  on  the  arm 
OA. 


FIG.  150. 

The  wheel  is  prevented  from  turning  backward  by  the  pawl 
with  a  center  at  B. 

The  wheel  teeth  and  pawl  should  be  so  shaped  that  when  a 
load  is  applied,  the  pawl  will  not  tend  to  leave  contact  with  the 
wheel  tooth.  To  obtain  this  result  a  common  normal  to  the  front 
face  of  the  pawl  and  teeth  should  pass  between  the  pawl  and 
ratchet  wheel  centers.  CD  is  such  a  normal.  If  the  teeth  and 
pawl  were  so  shaped  that  the  common  normal  were  CD',  the 
pawl  would  then  tend  to  disengage  itself  from  the  wheel  when 
under  load. 

153.  In  using  the  pawl  with  center  at  B,  to  prevent  backward 
motion  of  the  wheel,  it  will  only  do  it  after  the  wheel  has  rotated 

155 


156 


MECHANISM 


backward  far  enough  for  the  pawl  to  engage  the  teeth,  which 
may  vary  from  zero  to  the  pitch  of  the  teeth. 

This  backward  motion  can  be  reduced  by  making  the  pitch 
smaller,  which  weakens  the  teeth,  and  is  not  always  desirable. 
It  can  also  be  reduced  by  placing  several  pawls  side  by  side  on 
the  pin  and  making  them  of  different  lengths.  In  this  way  the 
backward  motion  can  be  reduced  to  the  pitch  divided  by  the 
member  of  pawls. 


FIG.  151. 

By  placing  a  number  of  driving  pawls  side  by  side  and  having 
them  of  different  lengths,  a  very  fine  feed  can  be  obtained  without 
decreasing  the  pitch.  An  illustration  of  this  is  the  set  works  on 
sawmill  log  carriages.  Here  a  fine  feed  is  required  for  sawing 
various  thickness  but  fine  teeth  would  not  stand  the  wear  and 
tear. 


.   .  FIG.  152. 

Fig.  151  shows  this  arrangement  with  two  pawls  in  which  one 
is  made  one-half  of  the  pitch  of  the  teeth  longer  than  the  other. 

Fig.  152  illustrates  a  simple  ratchet  wheel  and  pawl  as  applied 
to  the  Weston  Ratchet  or  Scotch  drill.  The  friction  of  the  drill 
in  the  hole  in  this  case  prevents  backward  motion.  Fig.  153 
shows  a  section  of  the  "Armstrong  Universal  Ratchet,"  in  which 


INTERMITTENT  MOTIONS 


157 


there  are  12  teeth  on  the  wheel  and  four  pawls,  which  engage 
one  at  a  time.  The  universal  quality  of  the  tool  as  claimed  by 
the  makers  is  due  only  to  the  fact  that  the  axis  of  the  two 
trunnions  on  which  the  handle  turns,  is  at  an  acute  angle  with 


FIG.  153. 

the  axis  of  the  drill,  so  that  a  very  small  motion  of  the  end  of 
the  handle  will  turn  the  drill. 

To  get  a  motion  of  the  wheel  in  one  direction  for  both  the 
forward  and  backward  motions  of  the  pawl  lever  the  methods  of 
Figs.  154  or  155  may  be  employed. 


FIGS.  154. 


FIG.  155. 


If  it  is  desired  to  get  a  motion  of  the  wheel  in  either  direction, 
the  form  of  reversing  pawl  shown  in  Fig.  156  may  be  used.  In 
this  case  it  will  be  noted  that  the  form  of  wheel  teeth  are  different 


158 


MECHANISM 


from  where  motion  in  one  direction  only  is  required.     This 
form  is  sometimes  used  on  machine  tool  feed  mechanisms. 

When  it  is  necessary  to  vary  the  motion  of  the  ratchet  wheel 
without  varying  that  of  the  pawl,  it  may  be  done  by  "masking" 
one  or  more  teeth  of  the  wheel. 


FIG.  156. 


In  Fig.  157,  if  the  full  movement  of  the  pawl  will  advance  the 
wheel,  say  eight  teeth,  then  the  cam  6,  which  has  a  slightly 
larger  diameter  than  the  outside  diameter  of  the  wheel,  is  cut 
away  on  one  side  so  that  the  pawl  can  act  on  the  wheel  tooth 


FIG.  157. 

during  the  whole  angle,  and  hole  number  1  is  located  in  the 
plate.  Then  the  cam  is  revolved  on  its  shaft  so  that  the 
pawl  will  act  on  the  teeth  during  the  last  seven-eights  of  its 
motion,  and  hole  number  2  is  located,  and  so  on,  locating  the 


INTERMITTENT  MOTIONS 


159 


FIG.  158. 


FIG.  159. 


FIG.  160. 


160 


MECHANISM 


holes  so  that  one  tooth  will  be  cut  out  each  time,  till  in  the  ninth 
hole  all  of  the  teeth  are  masked.  The  width  of  pawl  face  must 
be  equal  to  the  face  of  ratchet  wheel  plus  the  cam  face.  A 
mechanism  similar  to  this  is  used  on  some  of  the  positive  feed 
engine  lubricators  and  on  printing  press  inking  mechanisms. 

154.  It  is  not  always  necessary  that  the  connection  between 
the  wheel  and  pawl  be  of  positive  form  as  shown  in  the  foregoing 
figures,  and  in  some  cases  a  friction  drive  may  be  substituted  as 
in  Figs.  158,  159  or  160.  These  are  sometimes  called  silent 
ratchets. 


FIG.  161. 


FIG.  162. 


155.  The  ratchet  mechanisms  so  far  discussed  together  with 
many  variations  of  them  are  suitable  for  feed  mechanisms 
where  the  motions  of  the  driver  are  not  too  rapid;  in  such  cases 
the  shock  at  the  beginning  of  motion  may  be  too  great,  and  the 
inertia  of  the  wheel  may  be  such  as  to  cause  it  to  "over-travel." 
In  such  mechanisms  as  revolution  counters  where  a  definite 
motion  of  the  follower  must  be  secured  each  time,  some  means 
must  be  employed  to  prevent  over-travel.  A  device  for  this 
purpose  is  shown  in  Fig.  161.  The  lever  to  which  the  pawl  is 
attached  has  a  projecting  beak  so  formed  that  when  the  pawl 
first  acts  on  a  pin,  the  end  C  of  this  beak  passes  across  the  line  of 
motion  of  the  pins  and  limits  their  motion.  The  curve  CD  is  an 
arc  with  0  as  a  center. 

The  magnetic  ratchet  .shown  in  Fig.  162  also  accomplishes  the 
same  result.  This  consists  of  a  metal  disk  A  and  an  arm  B  that 


INTERMITTENT  MOTIONS 


161 


swings  about  the  center  of  A.  On  B  is  a  magnet  C,  which  grips 
the  plate  and  by  a  commutator  device  is  released  for  the  return 
stroke  of  the  lever. 


FIG.  163. 


FIG.  164. 

As  soon  as  the  magnet  C  releases  A  a  second  magnet  on  the 
back  (not  shown),  grips  it  and  prevents  any  backward  movement 
of  the  disk.  These  ratchets  are  used  on  dividing  engines. 

156.  In  all  of  the  ratchet  mechanisms  so  far  discussed,  a 


162 


MECHANISM 


rotary  motion  was  given  to  the  wheel,  but  ratchet  mechanisms 
can  be  used  to  impart  a  reciprocating  motion.  Two  examples  of 
this  kind  are  shown  in  the  lifting  jacks  of  Figs.  163  and  164.  In 
each  case,  the  pawl  a  does  the  lifting,  and  6  holds  the  load  while  a 
new  grip  is  taken  with  a.  Fig.  164  is  of  the  silent  ratchet  type. 


CLUTCHES 

157.  In  Fig.  165  is  shown  a  method  of  connecting  the  ends  of 
two  shafts,  A  and  B  that  are  in  alignment  so  that  motion  can  be 


FIG.  165. 

transmitted  from  B  to  A  in  one  direction  only.  This  is  accom- 
plished by  having  the  part  of  the  clutch  on  B  rigidly  keyed  to  it, 
and  the  part  on  A  fastened  by  a  feather  key  to  prevent  its  turn- 
ing, but  allowing  it  to  be  moved  along  the  shaft,  to  engage  with 
the  part  on  B.  These  jaws  can  be  made  the  other  hand  from 
that  shown  so  that  A  will  be  turned  in  the  opposite  direction. 


FIG.  166. 

If  it  is  desired  to  have  A  turn  in  either  direction  the  jaws  can 
be  made  square  as  shown  in  Fig.  166.  These  are  positive 
clutches. 

Another  type  of  clutch  is  that  in  which  the  motion  is  imparted 
from  one  part  to  the  other  by  means  of  friction  between  the 
surfaces. 

One  of  the  most  common  forms  of  this  type  is  shown  in  Fig. 
167  and  is  known  as  a  cone  clutch.  Another  form  is  shown  in 
Fig.  168,  in  which  the  friction  surfaces  are  alternate  disks  of 


INTERMITTENT  MOTIONS 


163 


different  materials  and  are  brought  into  contact  by  means  of  the 
toggle  joint  arrangement  K.L.M. 

In  the  figure  this  clutch  is  shown  on  a  single  shaft.     When 
the  clutch  is  "thrown  out"  the  part  A  which  has  a  pulley  fastened 


FIG.  167. 


FIG.  168. 

on  its  long  hub,  but  not  shown  in  the  figure,  does  not  revolve, 
but  is  made  to  revolve  by  the  action  of  the  toggle  joint,  bringing 
the  friction  surfaces  into  contact. 


INDEX 


Action,  angle  of,  85 

arc  of,  85 

line  of,  87 

Accelerated  motion,  52 
Addendum  circle,  83 
Angle  of  obliquity,  87 
Angular  velocity,  4,  5 
Annular  gear,  91,  104,  105,  109 
Approach,  angle  of,  85 
Approximate    methods    laying    out 

gear  teeth,  111,  122 
Armstrong  ratchet,  156 
Axis,  instantaneous,  11 

B 

Back  cones,  122 

gears,  138 

lash,  84 

Base  circle,  48,  87,  90,  92 
Belt,  crossed,  142 

length  of,  146 

for  intersecting  shafts,  146 

open,  142 

quarter  turn,  146 

shifter  cam,  74 
Belting,  142 
Bevel  cones,  79 

gears,  shop  drawing  of,  123 
Bilgram  bevel  gear  planer,  127 
Bricard's  straight  line  motion,  46 


Cam  base  circle,  48 

contact  with  follower,  47 
curves,  motions  used,  49 
constant  breadth,  65 

diameter,  64 
cylindrical,  66 
definition  of,  47 
"dwell"  or  "rest,"  51,  52 


Cam,  inverse,  70 
involute,  57  ' 
main  and  return,  64 
positive  motion,  63 
with  flat-face  follower,  48,  56 
with  off-set  follower,  54 
with  oscillating  follower,  58 
with  roll  follower,  48 
theoretical  curve,  50 
working  curve,  50 

Center,  instantaneous,  11 

Centre,  definition  of,  11 

location  of,  in  single  body,  12 

Centros,  location  of,  in  three  bodies 
moving  relatively  to  each 
other,  12 

Chain  drive,  151 

kinematic,  definition  of,  14 
Morse  rocker  joint,  154 
Renold  silent  chain,  153 

Circle,  addendum,  definition  of,  83 
dedendum,  definition  of,  83 
rolling,  98,  101,  105,  106,  108 
working  depth,  definition  of,  83 

Circular  pitch,  definition  of,  84,  131 

Clearance,  83 

Clutch,  cone,  162 

Clutches,  162 

Comparison  of  involute  and  cycloidal 
systems,  107 

Cones,  Evans  friction,  77 
stepped,  148 

Conjugate  methods  of  cutting  gear 
teeth,  115 

Counter  shaft,  142 

Crossed  belt,  142 

Crown  gears,  122 

Crowning  pulleys,  143 

Cutting  gear  teeth,  114,  115,  133 

Cycloid,  construction  of,  98 

Cycloidal  gears,  interchangeable,  106 
system  of  gear  teeth,  98 

165 


166 


INDEX 


Dedendum,  circle,  83 

Detent,  155 

Drafting  board  parallel  motion 

mechanisms,  39 
Drill,  Weston  or  Scotch,  158 
"Dwell,"  51,  52 

E 

Effect  of  changing  distance  between 
centers  of  involute  gears, 
92 

Elliptical  gears,  133 

Epicycloid,  construction  of,  99 

Evans  friction  cones,  77 


Face  of  tooth,  83 
Fellows  gear  shaper,  117 
Flank  of  tooth,  83 
Flat-face  follower,  48,  56 
Friction  gears,  76 

bevel  cones,  79 

brush  plate  and  wheel,  80 

Evans  cones,  78 

grooved  cylinders,  77 

plain  cylinders,  76 

Sellers  feed  disks,  78 

G 

Gear  cutters,  interchangeable,  table 

of,  114 

cutting,  114,  115,  133 
hobbing,  118 
shaper,  Fellows,  117 
teeth,  approximate  methods  of 

laying  out,  111,  112,  113 
limiting  length  of,  88,  90,  91 
standard  sizes  of,  93 
with  straight  flanks,  104 
train,  definition  of,  136 

direction  of  rotation  in,  137 
thread  cutting,  138 
value  of,  136 
Gearing,  ratchet,  155 
Gears,  back,  138 
bevel,  121 

approximate  methods  of  lay- 
ing out,  122 


Gears,  approximate  methods  of  cut- 
ting teeth,  126 

Bilgram  planer  for,  127 

shop  drawing  of,  123 

to  calculate  outside  diameter 

of,  126 
crown,  122 

comparison    of    involute     and 
cycloidal  systems  of  teeth,  107 
cycloidal     annular     gear     and 
pinion,  104 

interchangeable,  106 

spur,  101 

rack  and  pinion,  103 

system,  98 
elliptical,  133 
friction,  76 
helical,  109 
herringbone,  111 
idle,  137 
involute,  annular  and  pinion,  91 

interchangeable,  93 

rack  and  pinion,  89 

spur,  87,  96 

stub  tooth,  88 

system,  86 
miter,  122 
pin,  108 
toothed,  81 
tumbling,  139 
worm  and  worm  wheel,  130 

cutting,  133 
Grants'  odontograph,  112 

H 

Harmonic  motion,  51 
Helical  gears,  109 

motion,  3 
Helix  angle,  118 
Herringbone  gears,  111 
Hobbing  gears,  118 
Hypocycloid,  construction  of,  101 


Idle  gear,  137 

Indicator  motion,  Tabor,  8 

Thompson,  43 
Instantaneous  axis,  11 

center,  definition  of,  11 


INDEX 


167 


Instaneous,  location  of,  11 

motion,  11 
Interchangeable  gear  cutters,  table 

of,  114 

Inverse  cams,  70 
Involute  cam,  57 

gear  teeth,  86 

interference  in,  92 

gears,  interchangeable,  93 

method  of  constructing,  57 


Jack  shaft,  142 


K 


Kinematic  chain,  14 

location  of  centros  in,  15 


Lantern  wheel,  109 

Length  of  belts,  146 
Linear  velocity,  4,  5,  11,  18,  25 
Loose  pulley,  144 

M 

Machine,  definition  of,  1 

design  of,  1 
Magnetic  ratchet,  160 
Main  and  return  cams,  64 
Mechanisms,  definition  of,  10 

modes  of  connection  of,  10 

parallel  motion,  37 

straight   line  motion,  41 

variable  motion,  29 
Miter  gears,  122 
Molding     generating     principle     of 

cutting  gear  teeth,  116 
Morse  rocker  joint  chain,  154 
Motion,  definition  of,  1 

forms  of,  2 

helical,  3 

plane,  2 

spherical,  3 

harmonic,  51 

instantaneous,  11 

parallel,  37 

quick  return,  134 
oscillating  arm,  34 
Whit  worth,  31,  129 


Motion,  straight  line,  41 

Bricard's,  46 

Peaucellius',  45 

Roberts,  45 

Scott  Russell,  42 

Tchebicheff,  44 

Thompson  indicator,  43 

uniform,  49 

uniformly  accelerated,  52 

retarded,  54 
Motions,  intermittent,  155 


0 


Obliquity,  angle  of,  87 

line  of,  90,  91 

Octoid  gear  tooth,  122,  128 
Odontograph,  Grant,  112 

Robinson,  111 

Willis,  112 
Off-set  follower,  54 
Open  belt,  142 
Oscillating  follower,  58 


Parallel  motion  mechanisms,  defini- 
tion of,  37 

drafting  board,  39 

ruler,  39 

Roberval  balance,  39 

parallelogram,  37 
Path,  definition  of,  3 
Pawl,  155,  157 
Pin  gears,  108 
Pinion  for  annular  gear,  limiting  size 

of,  105 
Pitch,  84,  131 

arc,  89 

circle,  82 

circular,  84,  131 

diameter,  84 

diametral,  84 

cones,  121 

point,  82 

Positive  motion  cam,  63 
Problems,  7,  17,  23,  35,  46,  62,  73, 

85,  97,  107,  129,  134,  140 
Pulleys,  crowning,  143 

tight  and  loose,  144 


168 


INbEX 


Q 

Quarter  turn  belts,  146 
Quick  return  motion,  134 

oscillating  arm,  34 

Whitworth,  31,  129 

R 

Rack  and  pinion,  cycloidal  system, 

103 

involute  system,  89 
Radian,  5 
Radius  arm,  31 

center,  31 

Ratchet  gearing,  155 
Ratchets,  silent,  160 

magnetic,  160 
Recess,  angle  of,  85 
Retarded  motion,  54 
Reversing  pawl,  157 
Renold  silent  chain,  153 
Roberts  straight  line  motion,  45 
Roberval  balance,  39 
Robinson  odontograph,  111 
Roll  follower,  48,  59,  64,  69,  73 
Rolls,  friction,  76 

grooved  friction,  77 

velocity,  ratio  of,  76 
Root  circle,  83 
Rope  drive,  150 
Rotation,  3 

S 

Screw  cutting  gear  train,  138 

Sellers  feed  disks,  78 

Shop  drawing  of  bevel  gears,  123 

Silent  ratchets,  160 

Spherical  motion,  3 

Sprocket  wheel,  151 

Spur  gears,  87,  96,  101 

Stepped  cones,  148 

gears,  109 
Straight  line  motions,  Bricard's,  46 

Peaucellier's,  45 

Roberts,  45 

Scott  Russell,  42 

Tchebicheff's,  44 

Thompson  indicator,  43 

Watt's,  41 


Tables     of     interchangeable     gear 
cutters,  115 

of  standard  tooth  parts,  94,  95 
Tchebicheff's  straight  line  motion, 

44 

Theoretical  cam  curve,  50 
Thompson  indicator  motion,  43 
Thread,  multiple,  130 

cutting,  138 
Throat  diameter,  132 
Tight  pulleys,  144 
Tooth  parts,  definition  of,  83 

tables  of,  94,  95 
Translation,  3 
Tredgold's  method,  122 
Tumbler  gears,  139 
Twisted  gears,  110 

U 

Uniform  motion,  49 
Uniformly  accelerated  motion,  52 
retarded  motion,  54 


Variable  motion  mechanisms,  29 
Velocity,  angular,  4,  5 

definition  of,  3 

diagrams,  25 

linear,  4,  5,  11,  18,  25 
belting,  142 
bevel  gears,  79,  122 
brush  plate  and  wheel,  80 
effect  of  idle  wheel  on,  137 
elliptical  gears,  134 
Evans  friction  cones,  78 
gearing,  82 
gear  trains,  136 
rolling  cylinders,  76 
Sellers  feed  disks,  79 
stepped  cones,  148 
worm  and  worm  wheel,  130 

relation    between    linear    an4 
angular,  5 

relative     linear,     solution     of 
problems  in,  18 

variable  linear,  4. 


INDEX  169 

W  Willis  odontograph,  112 

_T_  .  Working  cam  curve,  50 

Watts  straight  line  motion,  41  depth  g3 

Weston  ratchet,  156  w  ng   13Q 

Wheel,  lantern,  109  length  of,  132 

ratchet,  155  and 


_  151  method 

Whit  worth,  quick  return  motion,  31,  pitch    sd   131 

129  P 


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i3V  30183' 

NOV  2  IftibM 

OCT  IftinAi  . 

*°  'W  ftf. 

°fC  9  I*,  * 

"***  M 
opn     q  1942 

OCT  24   1946 

> 

LD  21-100m-7,'39(402s; 

YC   12712 


t 


if.  i  a  'Jo 


rfr 


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UNIVERSITY  OF  CALIFORNIA  LIBRARY 

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